5
The CHAPTER Derivative in Graphing and Applications
EXERCISE SET 5.1
1. (a) f > 0 and f > 0
y
(b) f > 0 and f < 0
y
x
x
(c) f < 0 and f > 0
y
(d) f < 0 and f < 0
y
x x
2. (a)
y
(b)
y
x
x
(c)
y
(d)
y
x
x
3.
A: dy/dx < 0, d2 y/dx2 > 0 B: dy/dx > 0, d2 y/dx2 < 0 C: dy/dx < 0, d2 y/dx2 < 0
4.
A: dy/dx < 0, d2 y/dx2 < 0 B: dy/dx < 0, d2 y/dx2 > 0 C: dy/dx > 0, d2 y/dx2 < 0
5. An inflection point occurs when f changes sign: at x = -1, 0, 1 and 2. 6. (a) f (0) < f (1) since f > 0 on (0, 1). (c) f (0) > 0 by inspection. (e) f (0) < 0 since f is decreasing there. 7. (a) [4, 6] (d) (2, 3) and (5, 7) (b) f (1) > f (2) since f < 0 on (1, 2). (d) f (1) = 0 by inspection. (f ) f (2) = 0 since f has a minimum there. (c) (1, 2) and (3, 5)
(b) [1, 4] and [6, 7] (e) x = 2, 3, 5
153
154
Chapter 5
8. f f
(1, 2) (2, 3) (3, 4) (4, 5) (5, 6) (6, 7) - - - + + - + - + + - - (b) f is decreasing on (-, 1], [3, +] (d) f is concave down on (2, 4) (b) f is nowhere decreasing (d) f is concave down on (1, 3) (b) (-, 3/2] (d) nowhere
9. (a) f is increasing on [1, 3] (c) f is concave up on (-, 2), (4, +) (e) points of inflection at x = 2, 4 10. (a) f is increasing on (-, +) (c) f is concave up on (-, 1), (3, +) (e) f has points of inflection at x = 1, 3 11. f (x) = 2(x - 3/2) f (x) = 2 (a) [3/2, +) (c) (-, +) (e) none (a) (-, -2] (c) nowhere (e) none (a) (-, +) (c) (-1/2, +) (e) -1/2 (a) [-2, 2] (c) (-, 0) (e) 0
12. f (x) = -2(2 + x) f (x) = -2
(b) [-2, +) (d) (-, +)
13. f (x) = 6(2x + 1)2 f (x) = 24(2x + 1)
(b) nowhere (d) (-, -1/2)
14. f (x) = 3(4 - x2 ) f (x) = -6x
(b) (-, -2], [2, +) (d) (0, +)
15. f (x) = 12x2 (x - 1) f (x) = 36x(x - 2/3)
(a) [1, +) (c) (-, 0), (2/3, +) (e) 0, 2/3 (a) [0, +), (c) (-, 1), (3/2, +) (e) 1, 3/2 (a) [ 3-2 5 , 3+2 5 ] (c)
(b) (-, 1] (d) (0, 2/3)
16. f (x) = x(4x2 - 15x + 18) f (x) = 6(x - 1)(2x - 3) 3(x2 - 3x + 1) (x2 - x + 1)3
2
(b) (-, 0] (d) (1, 3/2)
17. f (x) = - f (x) =
6x(2x - 8x + 5) (x2 - x + 1)4 x2 - 2 (x + 2)2
(0, 2 - 26 ), (2 + 26 , +) (e) 0, 2 - 6/2, 2 + 6/2 (a) (-, - 2), ( 2, +) (c) (-, - 6), (0, 6) (e) none (a) [-1/2, +) (c) (-2, 1) (e) -2, 1
(b) (-, 3-2 5 ], [ 3+2 5 , +) (d) (-, 0), (2 -
6 2 ,2
+
6 2 )
18. f (x) = - f (x) = 19. f (x) =
2x(x2 - 6) (x + 2)3 2x + 1 3(x2 + x + 1)2/3 2(x + 2)(x - 1) 9(x2 + x + 1)5/3
(b) (- 2, 2) (d) (- 6, 0), ( 6, +)
(b) (-, -1/2] (d) (-, -2), (1, +)
f (x) = -
Exercise Set 5.1
155
20. f (x) = f (x) =
4(x - 1/4) 3x2/3 4(x + 1/2) 9x5/3 4(x2/3 - 1) 3x1/3 4(x5/3 + x) 9x7/3
(a) [1/4, +) (c) (-, -1/2), (0, +) (e) -1/2, 0
(b) (-, 1/4] (d) (-1/2, 0)
21. f (x) = f (x) =
(a) [-1, 0], [1, +) (c) (-, 0), (0, +) (e) none
(b) (-, -1], [0, 1] (d) nowhere
22. f (x) =
2 -1/3 x -1 3 2 f (x) = - 4/3 9x
2
(a) [-1, 0], [1, +) (c) (-, 0), (0, +) (e) none (a) (-, 0] (c) (-, -1), (1, +) (e) -1, 1 (a) (-, +) (c) (0, +) (e) 0
(b) (-, -1], [0, 1] (d) nowhere
23. f (x) = -xe-x
/2 2 -x2 /2
f (x) = (-1 + x )e
(b) [0, +) (d) (-1, 1)
24. f (x) = (2x2 + 1)ex
2
2
f (x) = 2x(2x + 3)e
x2
(b) none (d) (-, 0)
25. f (x) =
x x2 + 4 x2 - 4 (x2 + 4)2
(a) [0, +) (c) (-2, +2) (e) -2, +2 (a) [e-1/3 , +) (c) (e
-5/6
(b) (-, 0] (d) (-, -2), (2, +)
f (x) = -
26. f (x) = x2 (1 + 3 ln x) f (x) = x(5 + 6 ln x)
(b) (0, e-1/3 ] (d) (0, e-5/6 )
, +)
(e) e-5/6 27. f (x) = 2x 1 + (x2 - 1)2 3x4 - 2x2 - 2 [1 + (x2 - 1)2 ]2 (a) [0 + ) (b) (-, 0] 1+ 1- 1- 7 (c) - 3 7 , - 3 7 , , 1+ 7 3 3 1- 1+ 7 (d) - , - 1+ 7 , - 3 7 , 1- 7 , , + 3 3 3 1 7 (e) four: 3 (a) [0, 1] (b) [-1, 0]
3/2
f (x) = -2
28. f (x) = f (x) =
3x1/3
2 1 - x4/3
2 -1 + 3x4/3 9x4/3 1 - x4/3
(c) (-1, -3-3/4 ), (0, 3-3/4 , 1) (d) (3-3/4 , 0), (3-3/4 (e) 3-3/4
156
Chapter 5
29. f (x) = cos x + sin x f (x) = - sin x + cos x (a) [-/4, 3/4] (c) (-3/4, /4) (e) -3/4, /4 (b) (-, -/4], [3/4, ) (d) (-, -3/4), (/4, )
^
1.5
6
1.5
30. f (x) = (2 tan2 x + 1) sec x f (x) = sec x tan x(6 tan2 x + 5) (a) [-/2, /2] (c) (0, /2) (e) 0 (b) nowhere (d) (-/2, 0)
^
10
6
10
1 31. f (x) = - sec2 (x/2) 2 1 f (x) = - tan(x/2) sec2 (x/2)) 2 (a) nowhere (b) (-, ) (c) (-, 0) (d) (0, ) (e) 0
10
C
c
10
32. f (x) = 2 - csc2 x
2
cos x f (x) = 2 csc x cot x = 2 3 sin x (a) [/4, 3/4] (b) (0, /4], [3/4, ) (c) (0, /2) (d) (/2, ) (e) /2
8
0 2
p
33. f (x) = 1 + sin 2x f (x) = 2 cos 2x f (x) = -4 sin 2x (a) (b) (c) (d) (e) [-, -3/4], [-/4, /4], [3/4, ] [-3/4, -/4], [/4, 3/4] (-/2, 0), (/2, ) (-, -/2), (0, /2) -/2, 0, /2
2
C 0
c
34. f (x) = 2 sin 4x f (x) = 8 cos 4x (a) (b) (c) (d) (e) (0, /4], [/2, 3/4] [/4, /2], [3/4, ] (0, /8), (3/8, 5/8), (7/8, ) (/8, 3/8), (5/8, 7/8) /8, 3/8, 5/8, 7/8
1
0 0
p
Exercise Set 5.1
157
35. (a)
4
y
(b)
4
y
(c)
4
y
x 2 2
x 2
x
36. (a)
4
y
(b)
4
y
(c)
4
y
x 2 2
x 2
x
37. (a) g(x) has no zeros: There can be no zero of g(x) on the interval - < x < 0 because if there were, say g(x0 ) = 0 where x0 < 0, then g (x) would have to be positive between x = x0 and x = 0, say g (x1 ) > 0 where x0 < x1 < 0. But then g (x) cannot be concave up on the interval (x1 , 0), a contradiction. There can be no zero of g(x) on 0 < x < 4 because g(x) is concave up for 0 < x < 4 and thus 2 the graph of g(x), for 0 < x < 4, must lie above the line y = - x + 2, which is the tangent 3 line to the curve at (0, 2), and above the line y = 3(x - 4) + 3 = 3x - 9 also for 0 < x < 4 (see figure). The first condition says that g(x) could only be zero for x > 3 and the second condition says that g(x) could only be zero for x < 3, thus g(x) has no zeros for 0 < x < 4. Finally, if 4 < x < +, g(x) could only have a zero if g (x) were negative somewhere for x > 4, and since g (x) is decreasing there we would ultimately have g(x) < -10, a contradiction. (b) one, between 0 and 4 y (c) We must have lim g (x) = 0; if the limit were -5
x+
then g(x) would at some time cross the line g(x) = -10; if the limit were 5 then, since g is concave down for x > 4 and g (4) = 3, g must decrease for x > 4 and thus the limit would be 4. 38. (a) f (x) = 3(x - a)2 , f (x) = 6(x - a); inflection point is (a, 0) (b) f (x) = 4(x - a)3 , f (x) = 12(x - a)2 ; no inflection points
4 slope 2/3 1 slope 4 4 x
39. For n 2, f (x) = n(n - 1)(x - a)n-2 ; there is a sign change of f (point of inflection) at (a, 0) if and only if n is odd. For n = 1, y = x - a, so there is no point of inflection. 40. If t is in the interval (a, b) and t < x0 then, because f is increasing, then f (t) - f (x) f (t) - f (x) 0. Thus f (x0 ) = lim 0. tx t-x t-x f (t) - f (x) 0. If x0 < t t-x
158
Chapter 5
41. f (x) = 1/3 - 1/[3(1 + x)2/3 ] so f is increasing on [0, +) thus if x > 0, then f (x) > f (0) = 0, 1 + x/3 - 3 1 + x > 0, 3 1 + x < 1 + x/3.
2.5
0 0
10
42. f (x) = sec2 x - 1 so f is increasing on [0, /2) thus if 0 < x < /2, then f (x) > f (0) = 0, tan x - x > 0, x < tan x.
10
0 0
6
43. x sin x on [0, +): let f (x) = x - sin x. Then f (0) = 0 and f (x) = 1 - cos x 0, so f (x) is increasing on [0, +).
4
0 1
4
44. Let f (x) = 1 - x2 /2 - cos x for x 0. Then f (0) = 0 and f (x) = -x + sin x. By Exercise 43, f (x) 0 for x 0, so f (x) 0 for all x 0, that is, cos x 1 - x2 /2. 45. (a) Let f (x) = x - ln(x + 1) for x 0. Then f (0) = 0 and f (x) = 1 - 1/(x + 1) 0 for x 0, so f is increasing for x 0 and thus ln(x + 1) x for x 0. (b) Let g(x) = x - 1 x2 - ln(x + 1). Then g(0) = 0 and g (x) = 1 - x - 1/(x + 1) 0 for x 0 2 since 1 - x2 1. Thus g is decreasing and thus ln(x + 1) x - 1 x2 for x 0. 2 (c)
2 1.2
0 0
2
0 0
2
46. (a) Let h(x) = ex - 1 - x for x 0. Then h(0) = 0 and h (x) = ex - 1 0 for x 0, so h(x) is increasing. (b) Let h(x) = ex -1-x- 1 x2 . Then h(0) = 0 and h (x) = ex -1-x. By Part (a), ex -1-x 0 2 for x 0, so h(x) is increasing.
Exercise Set 5.1
159
(c)
6
6
0 0
2
0 0
2
47. Points of inflection at x = -2, +2. Concave up on (-5, -2) and (2, 5); concave down on (-2, 2). Increasing on [-3.5829, 0.2513] and [3.3316, 5], and decreasing on [-5, -3.5829] and [0.2513, 3.3316].
250
5
5
250
48. Points of inflection at x 1/ 3. Concave up = on [-5, -1/ 3] and [1/ 3, 5], and concave down on [-1/ 3, 1/ 3]. Increasing on [-5, 0] and decreasing on [0, 5].
1
5
5
2
49. f (x) = 2
90x3 - 81x2 - 585x + 397 . The denominator has complex roots, so is always positive; (3x2 - 5x + 8)3 hence the x-coordinates of the points of inflection of f (x) are the roots of the numerator (if it changes sign). A plot of the numerator over [-5, 5] shows roots lying in [-3, -2], [0, 1], and [2, 3]. To six decimal places the roots are x = -2.464202, 0.662597, 2.701605. 2x5 + 5x3 + 14x2 + 30x - 7 . Points of inflection will occur when the numerator changes (x2 + 1)5/2 sign, since the denominator is always positive. A plot of y = 2x5 + 5x3 + 14x2 + 30x - 7 shows that there is only one root and it lies in [0, 1]. To six decimal place the point of inflection is located at x = 0.210970.
50. f (x) =
51. f (x1 ) - f (x2 ) = x2 - x2 = (x1 + x2 )(x1 - x2 ) < 0 if x1 < x2 for x1 , x2 in [0, +), so f (x1 ) < f (x2 ) 1 2 and f is thus increasing. 52. f (x1 ) - f (x2 ) = 1 1 x2 - x1 - = > 0 if x1 < x2 for x1 , x2 in (0, +), so f (x1 ) > f (x2 ) and x1 x2 x1 x2 thus f is decreasing.
53. (a) If x1 < x2 where x1 and x2 are in I, then f (x1 ) < f (x2 ) and g(x1 ) < g(x2 ), so f (x1 ) + g(x1 ) < f (x2 ) + g(x2 ), (f + g)(x1 ) < (f + g)(x2 ). Thus f + g is increasing on I. (b) Case I: If f and g are 0 on I, and if x1 < x2 where x1 and x2 are in I, then 0 < f (x1 ) < f (x2 ) and 0 < g(x1 ) < g(x2 ), so f (x1 )g(x1 ) < f (x2 )g(x2 ), (f g)(x1 ) < (f g)(x2 ). Thus f g is increasing on I. Case II: If f and g are not necessarily positive on I then no conclusion can be drawn: for example, f (x) = g(x) = x are both increasing on (-, 0), but (f g)(x) = x2 is decreasing there.
160
Chapter 5
54. (a) f and g are increasing functions on the interval. By Exercise 53, f + g is increasing. (b) f and g are increasing functions on the interval. If f 0 and g 0 on the interval then by Exercise 53, f g is concave up on the interval. If the requirement of nonnegativity is removed, then the conclusion may not be true: let f (x) = g(x) = x2 on (-, 0). Each is concave up, 55. (a) f (x) = x, g(x) = 2x 56. (a) f (x) = ex , g(x) = e2x (b) f (x) = x, g(x) = x + 6 (b) f (x) = g(x) = x (c) f (x) = 2x, g(x) = x (c) f (x) = e2x , g(x) = ex
57. (a) f (x) = 6ax + 2b = 6a x + concavity at x = -
b b , f (x) = 0 when x = - . f changes its direction of 3a 3a
b b so - is an inflection point. 3a 3a
(b) If f (x) = ax3 + bx2 + cx + d has three x-intercepts, then it has three roots, say x1 , x2 and x3 , so we can write f (x) = a(x - x1 )(x - x2 )(x - x3 ) = ax3 + bx2 + cx + d, from which it 1 b = (x1 + x2 + x3 ), which is the average. follows that b = -a(x1 + x2 + x3 ). Thus - 3a 3 (c) f (x) = x(x2 - 3x + 2) = x(x - 1)(x - 2) so the intercepts are 0, 1, and 2 and the average is 1. f (x) = 6x - 6 = 6(x - 1) changes sign at x = 1. b 58. f (x) = 6x + 2b, so the point of inflection is at x = - . Thus an increase in b moves the point of 3 inflection to the left. 59. (a) Let x1 < x2 belong to (a, b). If both belong to (a, c] or both belong to [c, b) then we have f (x1 ) < f (x2 ) by hypothesis. So assume x1 < c < x2 . We know by hypothesis that f (x1 ) < f (c), and f (c) < f (x2 ). We conclude that f (x1 ) < f (x2 ). (b) Use the same argument as in Part (a), but with inequalities reversed. 60. By Theorem 5.1.2, f is increasing on any interval [(2n-1), 2(n+1)] (n = 0, 1, 2, ), because f (x) = 1 + cos x > 0 on ((2n - 1), (2n + 1)). By Exercise 59 (a) we can piece these intervals together to show that f (x) is increasing on (-, +). 61. By Theorem 5.1.2, f is decreasing on any interval [(2n + /2, 2(n + 1) + /2] (n = 0, 1, 2, ), because f (x) = - sin x + 1 < 0 on (2n + /2, 2(n + 1) + /2). By Exercise 59 (b) we can piece these intervals together to show that f (x) is decreasing on (-, +). 62. By zooming on the graph of y (t), maximum increase is at x = -0.577 and maximum decrease is at x = 0.577. 63.
2 infl pt 1 t 1 no infl pts t y
64.
2
y
Exercise Set 5.2
161
65.
4 3 2 1
y
66.
4 infl pts 3 2 1 x
y
infl pts
x
67. (a) y (t) =
LAke-kt LAk S, so y (0) = (1 + Ae-kt )2 (1 + A)2
(b) The rate of growth increases to its maximum, which occurs when y is halfway between 0 and 1 L, or when t = ln A; it then decreases back towards zero. k dy (c) From (2) one sees that is maximized when y lies half way between 0 and L, i.e. y = L/2. dt This follows since the right side of (2) is a parabola (with y as independent variable) with 1 y-intercepts y = 0, L. The value y = L/2 corresponds to t = ln A, from (4). k 68. Find t so that N (t) is maximum. The size of the population is increasing most rapidly when t = 8.4 years. 69. t = 7.67
1000
0 0
15
70. Since 0 < y < L the right-hand side of (3) of Example 9 can change sign only if the factor L - 2y L L changes sign, which it does when y = L/2, at which point we have = , 1 = Ae-kt , 2 1 + Ae-kt 1 t = ln A. k
EXERCISE SET 5.2
1. (a)
f (x) y
(b)
y
x
f (x) x
162
Chapter 5
(c)
y f (x) x
(d)
y
f (x)
x
2. (a)
y
(b)
x
y x
(c)
y
(d)
y x
x
3. (a) f (x) = 6x - 6 and f (x) = 6, with f (1) = 0. For the first derivative test, f < 0 for x < 1 and f > 0 for x > 1. For the second derivative test, f (1) > 0. (b) f (x) = 3x2 - 3 and f (x) = 6x. f (x) = 0 at x = 1. First derivative test: f > 0 for x < -1 and x > 1, and f < 0 for -1 < x < 1, so there is a relative maximum at x = -1, and a relative minimum at x = 1. Second derivative test: f < 0 at x = -1, a relative maximum; and f > 0 at x = 1, a relative minimum. 4. (a) f (x) = 2 sin x cos x = sin 2x (so f (0) = 0) and f (x) = 2 cos 2x. First derivative test: if x is near 0 then f < 0 for x < 0 and f > 0 for x > 0, so a relative minimum at x = 0. Second derivative test: f (0) = 2 > 0, so relative minimum at x = 0. (b) g (x) = 2 tan x sec2 x (so g (0) = 0) and g (x) = 2 sec2 x(sec2 x + 2 tan2 x). First derivative test: g < 0 for x < 0 and g > 0 for x > 0, so a relative minimum at x = 0. Second derivative test: g (0) = 2 > 0, relative minimum at x = 0. (c) Both functions are squares, and so are positive for values of x near zero; both functions are zero at x = 0, so that must be a relative minimum. 5. (a) f (x) = 4(x - 1)3 , g (x) = 3x2 - 6x + 3 so f (1) = g (1) = 0. (b) f (x) = 12(x - 1)2 , g (x) = 6x - 6, so f (1) = g (1) = 0, which yields no information. (c) f < 0 for x < 1 and f > 0 for x > 1, so there is a relative minimum at x = 1; g (x) = 3(x - 1)2 > 0 on both sides of x = 1, so there is no relative extremum at x = 1. 6. (a) f (x) = -5x4 , g (x) = 12x3 - 24x2 so f (0) = g (0) = 0. (b) f (x) = -20x3 , g (x) = 36x2 - 48x, so f (0) = g (0) = 0, which yields no information. (c) f < 0 on both sides of x = 0, so there is no relative extremum there; g (x) = 12x2 (x - 2) < 0 on both sides of x = 0 (for x near 0), so again there is no relative extremum there. 7. f (x) = 16x3 - 32x = 16x(x2 - 2), so x = 0, 2 are stationary points. 8. f (x) = 12x3 + 12 = 12(x + 1)(x2 - x + 1), so x = -1 is the stationary point. 9. f (x) = -x2 - 2x + 3 , so x = -3, 1 are the stationary points. (x2 + 3)2
Exercise Set 5.2
163
10. f (x) = - 11. f (x) =
x(x3 - 16) , so stationary points at x = 0, 24/3 . (x3 + 8)2 2x ; so x = 0 is the stationary point. - 25)2/3
3(x2
12. f (x) =
2x(4x - 3) , so x = 0, 3/4 are the stationary points. 3(x - 1)1/3 sin x, sin x 0 so f (x) = - sin x, sin x < 0 cos x, sin x > 0 and f (x) does not exist - cos x, sin x < 0
13. f (x) = | sin x| =
xn
when x = n, n = 0, 1, 2, (the points where sin x = 0) because lim f (x) = lim f (x) (see Theorem preceding Exercise 61, Section 3.3). Now f (x) = 0 when - +
xn
cos x = 0 provided sin x = 0 so x = /2 + n, n = 0, 1, 2, are stationary points.
14. When x > 0, f (x) = cos x, so x = (n + 1 ), n = 0, 1, 2, . . . are stationary points. 2 When x < 0, f (x) = - cos x, so x = (n + 1 ), n = -1, -2, -3, . . . are stationary points. 2 f is not differentiable at x = 0, so the latter is a critical point but not a stationary point. 15. (a) none (b) x = 1 because f changes sign from + to - there (c) none because f = 0 (never changes sign) 16. (a) x = 1 because f (x) changes sign from - to + there (b) x = 3 because f (x) changes sign from + to - there (c) x = 2 because f (x) changes sign there 17. (a) x = 2 because f (x) changes sign from - to + there. (b) x = 0 because f (x) changes sign from + to - there. (c) x = 1, 3 because f (x) changes sign at these points. 18. (a) x = 1 19. critical points x = 0, 51/3 :f : x = 0: neither x = 51/3 : relative minimum 20. critical points x = -3/2, 0, 3/2: f : x = -3/2: relative minimum; x = 0: relative maximum; x = 3/2: relative minimum 21. critical points x = 2/3: f : x = 2/3: relative maximum; 22. critical points x = 7: f : x = 7: relative maximum; - x = 7: relative minimum 23. critical points x = 0: f : x = 0: relative minimum; (b) x = 5
0 0+ + + 0 51/3
(c) x = -1, 0, 3
0 + + + 0 3/2 0
0 + + + 3/2
+ + + 0 2/3
+ + +0 0+ + + 7 7
0 + + + 0
164
Chapter 5
24. critical points x = 0, ln 3: f : x = 0: neither; x = ln 3: relative minimum 25. critical points x = -1, 1:f : x = -1: relative minimum; x = 1: relative maximum 26. critical points x = ln 2, ln 3: f : x = ln 2: relative maximum; x = ln 3: relative minimum
0 0+ + + 0 ln 3
0+ + + 0 1 1
+ + +0 0+ + + ln 2 ln 3
27. f (x) = 8 - 6x: critical point x = 4/3 f (4/3) = -6 : f has a maximum of 19/3 at x = 4/3 28. f (x) = 4x3 - 36x2 : critical points at x = 0, 9 f (0) = 0: Theorem 5.2.5 with m = 3: f has an inflection point at x = 0 f (9) > 0: f has a minimum of -2187 at x = 9 29. f (x) = 2 cos 2x: critical points at x = /4, 3/4 f (/4) = -4: f has a maximum of 1 at x = /4 f (3/4) = 4 : f has a minimum of 1 at x = 3/4 30. f (x) = (x - 2)ex : critical point at x = 2 f (2) = e2 : f has a minimum of -e2 at x = 2 31. f (x) = 4x3 - 12x2 + 8x: critical points at x = 0, 1, 2 relative minimum of 0 at x = 0 0 + + + 0 0 + + + relative maximum of 1 at x = 1 0 1 2 relative minimum of 0 at x = 2 32. f (x) = 4x3 - 36x2 + 96x - 64; critical points at x = 1, 4 f (1) = 36: f has a relative minimum of -27 at x = 1 f (4) = 0: Theorem 5.2.5 with m = 3: f has an inflection point at x = 4 33. f f f f 34. f f f f (x) = 5x4 + 8x3 + 3x2 : critical points at x = -3/5, -1, 0 (-3/5) = 18/25 : f has a relative minimum of -108/3125 at x = -3/5 (-1) = -2 : f has a relative maximum of 0 at x = -1 (0) = 0: Theorem 5.2.5 with m = 3: f has an inflection point at x = 0 (x) = 5x4 + 12x3 + 9x2 + 2x: critical points at x = -2/5, -1, 0 (-2/5) = -18/25 : f has a relative maximum of 108/3125 at x = -2/5 (-1) = 0: Theorem 5.2.5 with m = 3: f has an inflection point at x = -1 (0) = 2 : f has a relative maximum of 0 at x = 0
35. f (x) =
2(x1/3 + 1) : critical point at x = -1, 0 x1/3 2 f (-1) = - : f has a relative maximum of 1 at x = -1 3 f does not exist at x = 0. By inspection it is a relative minimum of 0.
2x2/3 + 1 36. f (x) = : no critical point except x = 0; since f is an odd function, x = 0 is an inflection x2/3 point for f .
Exercise Set 5.2
165
37. f (x) = - 38. f (x) = -
5 ; no extrema (x - 2)2 2x(x4 - 16) ; critical points x = -2, 0, 2 (x4 + 16)2
1 1 f (-2) = - ; f has a relative maximum of at x = -2 8 8 1 f (0) = ; f has a relative minimum of 0 at x = 0 8 1 1 f (2) = - ; f has a relative maximum of at x = 2 8 8 39. f (x) = 2x ; critical point at x = 0 2 + x2
f (0) = 1; f has a relative minimum of ln 2 at x = 0 40. f (x) = 3x2 ; critical points at x = 0, -21/3 : f : 2 + x2
0+ + + 0 + + + 2
3
f (0) = 0 no relative extrema; f has no limit at x = -21/3 41. f (x) = 2e2x - ex ; critical point x = - ln 2 f (- ln 2) = 1/2; relative minimum of -1/4 at x = - ln 2 42. f (x) = 2x(1 + x)e2x : critical point x = -1, 0 f (-1) = -2/e2 ; relative maximum of 1/e2 at x = -1 f (0) = 2: relative minimum of 0 at x = 0 43. f (x) = (3 - 2x) : critical points at x = 3/2, 0, 3 f (3/2) = -2, relative maximum of 9/4 at x = 3/2 x = 0, 3 are not stationary points: by first derivative test, x = 0 and x = 3 are each a minimum of 0
0
44. On each of the intervals (-, -1), (-1, +) the derivative is of the form y =
1 hence it is 3x2/3 1 clear that the only critical points are possibly -1 or 0. Near x = 0, x = 0, y = > 0 so y 3x2/3 has an inflection point at x = 0. At x = -1, y changes sign, thus the only extremum is a relative minimum of 0 at x = -1.
y
2
45.
46.
x
y 20
(4, 17)
2
1
3
5
10
(4 + 17, 0) (0, 1)
x 2 4 6 8
4 6
(4 17, 0)
(
3 2
,
25 4
)
166
Chapter 5
y
(2, 49)
47.
48.
6
7 + 57 4
y
(
(0, 5)
,0
)
x 1
4
(0, 2)
( 1 , 50 ) 3 27
(2, 0) x
(
7 57 4
,0
) (
1 2
2
4
0.5 2
1.5
, 27 2
)
(3, 76)
80
4
49.
5 3 1 1
y
(
1 + 3 2
,
9 + 63 4
)
,
5 4
50.
+ 2
y
(0, 5)
(
1 2
,
5 4
2
)
(
1
1 2
)
( 5, 0)
3
(1, 0) (1, 0)
1 x 1.5 2 4
( 3, 4) (3, 4)
x 0.5 1.5
(5, 0)
(
51.
1 3 2
,
9 63 4
)
52.
0.4
(0, 0)
y
1.5
y
1 ( 1 , 27 ) 3
x 0.6
0.5
0.1 x
1
16 ( 2 , 729 ) ( 4 , 0 ) 9 9
1 1
0.3
(
1 2
,
27 16
)
y
0.4 0.2 1
0.5
53.
(
15 45 , 125 5
)
5 5 165 , 125 x
54.
y 0.4
(
21 216 7 , 2401 7
)
(
)
(1, 0)
(0, 0)
x 0.4 1.2
(1, 0)
(
5
5
,
165 125
)
15 45 , 125 5
0.6
(
) (
21 7
0.4 ,
216 7 2401
)
Exercise Set 5.2
167
55. (a)
curve crosses x-axis at x = 0, 1, -1
y 4 2 2 1 2 4 6 1 x
x-
lim y = -, lim y = +;
x+
(b)
x
lim y = +;
y 0.2
curve never crosses x-axis
x 1 1
(c)
curve crosses x-axis at x = -1
y 0.4 0.2 x 1 0.2 1
x-
lim y = -, lim y = +;
x+
(d)
x
lim y = +;
y
curve crosses x-axis at x = 0, 1
0.4
x 1 1
56. (a)
a
y
y
y
b
x x a b
a
b
x
(b)
y
y x a b
(c)
y
x a b
x a b
57. f (x) = 2 cos 2x if sin 2x > 0, f (x) = -2 cos 2x if sin 2x < 0, f (x) does not exist when x = /2, , 3/2; critical numbers x = /4, 3/4, 5/4, 7/4, /2, , 3/2 relative minimum of 0 at x = /2, , 3/2; relative maximum of 1 at x = /4, 3/4, 5/4, 7/4
0
1
0
o
168
Chapter 5
58. f (x) = 3 + 2 cos x; critical numbers x = 5/6, 7/6 relative minimum of 7 3/6 - 1 at x = 7/6; relative maximum of 5 3/6 + 1 at x = 5/6
12
0 0
o
59. f (x) = - sin 2x; critical numbers x = /2, , 3/2 relative minimum of 0 at x = /2, 3/2; relative maximum of 1 at x =
1
60. f (x) = (2 cos x - 1)/(2 - cos x)2 ; critical numbers x = /3, 5/3 relative maximum of at x = /3, 3/3 relative minimum of - 3/3 at x = 5/3
0.8
0
o
0 0
o
0.8
61. f (x) = ln x + 1, f (x) = 1/x; f (1/e) = 0, f (1/e) > 0; relative minimum of -1/e at x = 1/e
2.5
0 0.5
2.5
62. f (x) = -2
ex - e-x = 0 when x = 0. (ex + e-x )2
1
By the first derivative test f (x) > 0 for x < 0 and f (x) < 0 for x > 0; relative maximum of 1 at x = 0
2 0
2
63. f (x) = 2x(1 - x)e-2x = 0 at x = 0, 1. f (x) = (4x2 - 8x + 2)e-2x ; f (0) > 0 and f (1) < 0, so a relative minimum of 0 at x = 0 and a relative maximum of 1/e2 at x = 1.
0.14
0.3 0
4
Exercise Set 5.2
169
64. f (x) = 10/x - 1 = 0 at x = 10; f (x) = -10/x2 < 0; relative maximum of 10(ln(10) - 1) 13.03 at x = 10
14
0 4
20
65. Relative minima at x = -3.58, 3.33; relative maximum at x = 0.25
250
66. Relative minimum at x = -0.84; relative maximum at x = 0.84
1.2
5
5
-6
6
250
1.2
67. Relative maximum at x = -0.27, relative minimum at x = 0.22
y 8 6 4 2 x 4 2 f '(x) f "(x)
68. Relative maximum at x = -1.11, relative minimum at x = 0.47 relative maximum at x = 2.04
y
0.4 0.2 5 3 1 f "(x) f '(x)
x
5
4x3 - sin 2x 69. f (x) = , 2 x4 + cos2 x 6x2 - cos 2x (4x3 - sin 2x)(4x3 - sin 2x) f (x) = - 4(x4 + cos2 x)3/2 x4 + cos2 x Relative minima at x 0.62, relative maximum at x = 0 70. Point of inflection at x = 0
y 0.2 f '(x) 0.1 f "(x) x 0.1 0.1
f '(x)
2
y
x 2 f''(x) 2 1
0.2
170
Chapter 5
71. (a) Let f (x) = x2 +
k k 2x3 - k , then f (x) = 2x - 2 = . f has a relative extremum when x x x2 3 3 3 2x - k = 0, so k = 2x = 2(3) = 54. x k - x2 . f has a relative extremum when k - x2 = 0, so , then f (x) = 2 x2 + k (x + k)2 k = x2 = 32 = 9. (b) x 0.6436, 0
(b) Let f (x) =
72. (a) relative minima at x 0.6436, relative maximum at x = 0
y 2 1.8 1.6 1.4 1.2 1 1.5 0.5 0.5 1.5
x
73. (a) (-2.2, 4), (2, 1.2), (4.2, 3) (b) f exists everywhere, so the critical numbers are when f = 0, i.e. when x = 2 or r(x) = 0, so x -5.1, -2, 0.2, 2. At x = -5.1 f changes sign from - to +, so minimum; at x = -2 f changes sign from + to -, so maximum; at x = 0.2 f doesn't change sign, so neither; at x = 2 f changes sign from - to +, so minimum. Finally, f (1) = (12 - 4)r (1) + 2r(1) -3(0.6) + 2(0.3) = -1.2. 74. g (x) exists everywhere, so the critical points are the points when g (x) = 0, or r(x) = x, so r(x) crosses the line y = x. From the graph it appears that this happens precisely when x = 0. 75. f (x) = 3ax2 + 2bx + c and f (x) has roots at x = 0, 1, so f (x) must be of the form f (x) = 3ax(x - 1); thus c = 0 and 2b = -3a, b = -3a/2. f (x) = 6ax + 2b = 6ax - 3a, so f (0) > 0 and f (1) < 0 provided a < 0. Finally f (0) = d, so d = 0; and f (1) = a + b + c + d = a + b = -a/2 so a = -2. Thus f (x) = -2x3 + 3x2 . 76. (a) one relative maximum, located at x = n (b) f (x) = cxn-1 (-x + n)e-x = 0 at x = n. Since f (x) > 0 for x < n and f (x) < 0 for x > n it's a maximum.
0.3
0 0
14
77. (a) f (x) = -xf (x). Since f (x) is always positive, f (x) = 0 at x = 0, f (x) > 0 for x < 0 and f (x) < 0 for x > 0, so x = 0 is a maximum.
(b)
1 2c
y
( ,
1 2c
)
x
Exercise Set 5.3
171
78. (a) Because h and g have relative maxima at x0 , h(x) h(x0 ) for all x in I1 and g(x) g(x0 ) for all x in I2 , where I1 and I2 are open intervals containing x0 . If x is in both I1 and I2 then both inequalities are true and by addition so is h(x) + g(x) h(x0 ) + g(x0 ) which shows that h + g has a relative maximum at x0 . (b) By counterexample; both h(x) = -x2 and g(x) = -2x2 have relative maxima at x = 0 but h(x) - g(x) = x2 has a relative minimum at x = 0 so in general h - g does not necessarily have a relative maximum at x0 . 79. (a)
y
(b)
y
(c)
y
(
x0
)
x
(
x0
)x
(
x0
)
x
f (x0 ) is not an extreme value.
f (x0 ) is a relative maximum.
f (x0 ) is a relative minimum.
EXERCISE SET 5.3
1. Vertical asymptote x = 4 horizontal asymptote y = -2
y
2
2. Vertical asymptotes x = 2 horizontal asymptote y = 0
y 8
x=4 x
6 8
6 4 2 4 x = 2 4 6 8 x=2 4 x
y = 2
4 6
3. Vertical asymptotes x = 2 horizontal asymptote y = 0
y
4
4. Vertical asymptotes x = 2 horizontal asymptote y = 1
y 8 4
x = 2
2
x
4 2 4 4
y=1 4 2 6 x=2 4
x
x=2 x = 2
172
Chapter 5
5. No vertical asymptotes horizontal asymptote y = 1
y
1.25 0.75
6. No vertical asymptotes horizontal asymptote y = 1
y=1
(
x
1 3
18 333 ,
((
1 3
18 3 33
3 33 / 3
) )
1
2
)
y
(0, 1)
y=1 0.8
4
( , ) ( , )
2 3 1 4 2 3 1 4
x 4 2 2 4
(1, 0) (1, 0)
(
7. Vertical asymptote x = 1 horizontal asymptote y = 1
y
4 2
1 3
18 + 333 ,
((
1 3
18 3 33
3 + 33 / 3
) )
1
2
)
8. Vertical asymptote x = 0, -3 horizontal asymptote y = 2
y 8
y=1
(2, )
7 4
y=2 x 2
x
2 2 4
6
4
2
2 6
x=1 x = 3
(
3
1 2
,1
3
)
9. Vertical asymptote x = 0 horizontal asymptote y = 3
y
10. Vertical asymptote x = 1 horizontal asymptote y = 3
y 18
8
12
4
(2, 3)
(6, )
25 9
(2, )
1 3
6
x=1 y=3 x 3 6
y=3
5
(4, )
11 4
x
5 (1, 0)
Exercise Set 5.3
173
11. Vertical asymptote x = 1 horizontal asymptote y = 9
y
30
12. Vertical asymptote x = 1 horizontal asymptote y = 3
y 15 12
( )
1 ,9 3
20
(
x=1 x
10
7 3
,
7473 2500
)
9 6
(
x=1 4
5 3
,
6117 2048
)
x
y = 10
y=3 10 6 2
10 (1, 1)
(
1 3,0
)
8 12
13. Vertical asymptote x = 1 horizontal asymptote y = -1
y
14. Vertical asymptote x = 1 horizontal asymptote y = 0
(
1, 1 2
)
((
x=1 x
2 4
28 + 12 5)1/3 2
, y
(28 + 12 5)2/3
18 + 6 5
)
)
6
5 x=1 3
y = 1
2 4
(2
4 2 2 4
1/3
, 23
2/3
x
4
15. (a) horizontal asymptote y = 3 as x , vertical asymptotes of x = 2
y 10
(b) horizontal asymptote of y = 1 as x , vertical asymptotes at x = 1
y 10
5 x x 5 5 5 10 5 5
174
Chapter 5
(c) horizontal asymptote of y = -1 as x , vertical asymptotes at x = -2, 1
y 10
(d) horizontal asymptote of y = 1 as x , vertical asymptote at x = -1, 2
y 10
5
5
x
x 5 5
10 10
16. (a)
y
x a b
(b) Symmetric about the line x = Note that
a+b means f 2
a+b +x 2
=f
a+b -x 2
for any x.
a+b a-b a+b a-b +x-a = x- and +x-b = x+ , and the same equations 2 2 2 2 are true with x replaced by -x. Hence a+b +x-a 2 a+b +x-b 2 = x- a-b 2 x+ a-b 2 = x2 - a-b 2
2
The right hand side remains the same if we replace x with -x, and so the same is true of the left hand side, and the same is therefore true of the reciprocal of the left hand side. But the reciprocal of the left hand side is equal to f a+b + x . Since this quantity remains 2 a+b has unchanged if we replace x with -x, the condition of symmetry about the line x = 2 been met. x2 9 - (x + 3) = lim =0 x x - 3 x-3
10
17.
x
lim
y
y=x+3
x
10 5
x=3
Exercise Set 5.3
175
18.
2 + 3x - x3 2 - (3 - x2 ) = 0 as x x x
y 25 15 5 5 3 1 10 20 y = 3 x2 1 3 5 x
19. y = x2 - y =
1 x3 - 1 = ; x x
20. y =
2x3 + 1 , x2 y = 0 when
3
x2 - 2 2 = x - so x x y = x is an oblique asymptote; y =
x=- y =
1 -0.8; 2
x2 + 2 , x2 4 y =- 3 x
y 4 y=x
2(x3 - 1) x3
y
x 4
(0.8, 1.9) (1, 0)
x
21. y =
(x - 2)3 12x - 8 =x-6+ so x2 x2 y = x - 6 is an oblique asymptote; y =
10 (4, 13.5)
y 10 (2, 0) x y =x6
(x - 2)2 (x + 4) , x3 24(x - 2) y = x4
22. y = x -
1 1 1 1 - 2 =x- + 2 x x x x y = x is an oblique asymptote; 1 1 y = 1 + 2 + 3, x x 1 1 y = -2 3 - 6 4 x x
so
6 4
(1, 1)
y y=x
x 2 4 6
6 4 2
(3, )
25 9
176
Chapter 5
23. y =
x3 - 4x - 8 8 = x2 - 2x - so x+2 x+2 y = x2 - 2x is a curvilinear asymptote as x
(3, 23) 4
y
30
y = x 2 2x
10
x
4 (0, 4)
x = 2
24. y =
x5 x = x3 - x + 2 so x2 + 1 x +1 y = x3 - x is a curvilinear asymptote
y 10 5
y = x3 x x 1 2
2
1 5 10
25. (a) VI
(b) I
(c) III
(d) V
(e) IV
(f ) II
26. (a) When n is even the function is defined only for x 0; as n increases the graph approaches the line y = 1 for x > 0.
y
x
(b) When n is odd the graph is symmetric with respect to the origin; as n increases the graph approaches the line y = 1 for x > 0 and the line y = -1 for x < 0.
y
x
Exercise Set 5.3
177
27. y =
4x2 - 1
28. y = y =
3
x2 - 4;
4x 4x2 - 1 4 y =- so 2 - 1)3/2 (4x y = 1 extrema when x = , 2 no inflection points
y
4
2x ; 3(x2 - 4)2/3 2(3x2 + 4) 9(x2 - 4)5/3
y 3
y =-
(2, 0) 2
(2, 0) 2 (0, 2)
x
3 2 1
x
1 1
29. y = 2x + 3x2/3 ; y = 2 + 2x-1/3 ; 2 y = - x-4/3 3
y 5
30. y = 2x2 - 3x4/3 y = 4x - 4x1/3 4 y = 4 - x-2/3 3
y 8 6
x (0, 0) 4
4 2 x 4 2
(1, 1)
2
(1, 1)
4
31. y = x1/3 (4 - x); y = 4(1 - x) ; 3x2/3 4(x + 2) y =- 9x5/3
10 y
32. y = 5x2/3 + x5/3 5(x + 2) 3x1/3 10(x - 1) y = 9x4/3 y =
y 8
(1, 3) x (2, 6 2 ) 10
6 4 2
3
6
(1, 6)
(2, 3 22/3)
4
x 2 2 4
178
Chapter 5
33. y = x2/3 - 2x1/3 + 4 y = 2(x1/3 - 1) 3x2/3 2(x1/3 - 2) 9x5/3
y
34. y =
y =-
8( x - 1) ; x 4(2 - x) y = ; x2 2(3 x - 8) y = x3
4 y (4, 2)
(8, 12)
(64 , 15 ) 9 8
x 15
6 (1, 3) 2 10
(0, 4) (8, 4)
x
10 30
35. y = x + sin x; y = 1 + cos x, y = 0 when x = + 2n; y = - sin x; y = 0 when x = n n = 0, 1, 2, . . .
y c c
36. y = x - tan x; y = 1 - sec2 x; y = 0 when x = 2n y = -2 sec2 x tan x = 0 when x = n n = 0, 1, 2, . . .
x = 4.5 10 6 y x = 4.5
x
2 5 3 4 8 x = 1.5 x = 1.5 3 5 x
37. y y y y y
= 3 cos x + sin x; = - 3 sin x + cos x; = 0 when x = /6 + n; = - 3 cos x - sin x; = 0 when x = 2/3 + n
y 2 o 2 x
38. y y y y y
= sin x + cos x; = cos x - sin x; = 0 when x = /4 + n; = - sin x - cos x; = 0 when x = 3/4 + n
2 y x
o 2
o
Exercise Set 5.3
179
39. y = sin2 x - cos x; y = sin x(2 cos x + 1); y = 0 when x = , 2, 3 and when 2 2 4 8 x = - , , , ; 3 3 3 3 y = 4 cos2 x + cos x - 2; y = 0 when x 2.57, 0.94, 3.71, 5.35, 7.22, 8.86
(2.57, 1.13) 5 5 (*, 4 ) (8, 4 ) (2.57, 1.13) (C, 1) 1.5 (c, 1) (3.71, 1.13) 5 5 (g, 4 ) (w, 4 ) (8.86, 1.13) (, 1)
40.
y=
tan x;
sec2 x y = ; so y > 0 always; 2 tan x y = sec2 x 3 tan2 x - 1 , 4(tan x)3/2 y = 0 when x = 6
y 10 8 6 4 2 3 x=6 x
y
x
4 (0.94, 0.06) 1 6 (0.94, 0.06) (5.35, 0.06) 10 (7.22, 0.06) (o, 1)
41. (a)
x+
lim xex = +, lim xex = 0
x-
y 1 5 3 x 1
(b) y = xex ; y = (x + 1)ex ; y = (x + 2)ex
(2, 0.27) (1, 0.37)
42.
x+
lim f (x) = 0, lim f (x) = -
x- -x -x
y 0.2
f (x) = (1 - x)e , f (x) = (x - 2)e critical point at x = 1; relative maximum at x = 1 point of inflection at x = 2 horizontal asymptote y = 0 as x +
1 (1, e )
(2, e22)
x 2
1
0.8
43. (a)
x2 x2 = 0, lim 2x = + x+ e2x x- e lim
0.3
y
(b) y = x2 /e2x = x2 e-2x ; y = 2x(1 - x)e-2x ; y = 2(2x - 4x + 1)e
2 2 -2x
;
y = 0 if 2x - 4x + 1 = 0, when 4 16 - 8 = 1 2/2 0.29, 1.71 x= 4
(1, 0.14) (1.71, 0.10) x 1 2 3 (0, 0) (0.29, 0.05)
180
Chapter 5
44. (a)
x+
lim x2 e2x = +, lim x2 e2x = 0.
x-
y 0.3 (1, 0.14) (1.71, 0.10) 0.2 x (0, 0)
(b) y = x2 e2x ; y = 2x(x + 1)e2x ; y = 2(2x2 + 4x + 1)e2x ; y = 0 if 2x2 + 4x + 1 = 0, when -4 16 - 8 = -1 2/2 -0.29, -1.71 x= 4 lim x2 e-x = 0
y
2
3 2 1 (0.29, 0.05)
45. (a) (b)
x
y = x2 e-x ;
2
(1, e )
1
(1, e )
1
y = 2x(1 - x2 )e-x ; y = 0 if x = 0, 1; y = 2(1 - 5x2 + 2x4 )e-x y = 0 if 2x4 - 5x2 + 1 = 0, 5 17 , x2 = 4 x = 1 5 + 17 1.51, 2 x = 1 5 - 17 0.47 2
y 1
2
2
(1.51, 0.23) (0.47, 0.18) 0.1
(1.51, 0.23) (0.47, 0.18)
x
3 1 1 3
46. (a)
x
lim f (x) = 1
2
(b) f (x) = 2x-3 e-1/x so f (x) < 0 for x < 0 and f (x) > 0 for x > 0. Set u = x2 and use the given result to find lim f (x) = 0, so (by the First Derivative Test) f (x)
x0
0.4
has a minimum at x = 0. f (x) = (-6x-4 + 4x-6 )e-1/x , so f (x) has points of inflection at x = 2/3.
2
( 2/3, e 3/2)
10 5
(2/3, e 3/2)
5 10
x
(0, 0)
47. (a)
x-
lim f (x) = 0, lim f (x) = -
x+
y
10
(b) f (x) = -
ex (x - 2) so (x - 1)2
1
x=1 x
2 10 3 4
f (x) = 0 when x = 2 f (x) = - ex (x2 - 4x + 5) so (x - 1)3
(2, e 2 )
f (x) = 0 always relative maximum when x = 2, no point of inflection asymptote x = 1
20
Exercise Set 5.3
181
y 3
48. (a)
x-
lim f (x) = 0, lim f (x) = +
x+
ex (3x + 2) (b) f (x) = so 3x1/3 2 f (x) = 0 when x = - 3 e (9x + 12x - 2) so 9x4/3 points of inflection when f (x) = 0 at 2- 6 2+ 6 ,- , x=- 3 3
x 2
2
(2/3, 2/32/3e 2/3) 1 (1.48, 0.30) (0.15, 0.33)
f (x) =
x
2
1
1
relative maximum at
2 2 - , e-2/3 - 3 3
2/3
absolute minimum at (0, 0)
49.
x+
lim f (x) = 0, lim f (x) = +
y 1.8
(2, 4 ) e
(3.41, 1.04) (0.59, 0.52) x 1 2 3 4
x-
f (x) = x(2 - x)e1-x , f (x) = (x2 - 4x + 2)e1-x critical points at x = 0, 2; relative minimum at x = 0, relative maximum at x = 2 points of inflection at x = 2 2 horizontal asymptote y = 0 as x + 50. lim f (x) = +, lim f (x) = 0
x- x-1
1 0.6
(0, 0)
x+
y 0.8 0.4 (0, 0) 4 (4.7, 0.35) 2 0.4 (3, 0.49)
f (x) = x2 (3 + x)e , f (x) = x(x2 + 6x + 6)ex-1 critical points at x = -3, 0; relative minimum at x = -3 points of inflection at x = 0, -3 3 0, -4.7, -1.27 horizontal asymptote y = 0 as x -
x 1 (1.27, 0.21)
51. (a)
x0+
lim y = lim x ln x = lim
x0+
x0+
ln x 1/x = lim = 0; 1/x x0+ -1/x2
y
x+
lim y = + = x ln x, = 1 + ln x, = 1/x, = 0 when x = e-1
x 1 (e 1, e 1)
(b) y y y y
182
Chapter 5
52. (a)
x0+
lim y = lim
x0+
ln x 1/x = lim = 0, 2 + -2/x3 1/x x0
0.2 0.1
y
x+
lim y = +
x 1
(b) y = x2 ln x, y = x(1 + 2 ln x), y = 3 + 2 ln x, y = 0 if x = e-1/2 , y = 0 if x = e-3/2 , lim y = 0
x0+
(e 3/2, 3e 3)0.1 2
0.2
(e 1/2, 1 e 1) 2
53. (a)
x0+
lim x2 ln x = 0 by the rule given, lim x2 ln x = + by inspection, and f (x) not defined
x+
for x < 0 (b) y y y y = x2 ln 2x, y = 2x ln 2x + x = 2 ln 2x + 3 = 0 if x = 1/(2 e), = 0 if x = 1/(2e3/2 )
y
3 (2e1 , 8e )
3/2 3 1 2
x
1 (21 e, 8e)
54. (a)
x+
lim f (x) = +; lim f (x) = 0
x0 2 2
y
(b) y = ln(x + 1), y = 2x/(x + 1) x2 - 1 y = -2 2 (x + 1)2 y = 0 if x = 0 y = 0 if x = 1
2 1 (1, ln 2) 2 (0, 0) (1, ln 2) x 2
55. (a)
(b) y = x ln x 2 ln x + 3 y = 3x1/3
x+ 2/3
lim f (x) = +, lim f (x) = 0
x0+
y
5
(e
3/2, 3e 2
)
3
3 y = 0 when ln x = - , x = e-3/2 2 -3 + 2 ln x y = , 9x4/3 3 y = 0 when ln x = , x = e3/2 2
1
x
1
(e
3/2,
3 2e
)
4
5
56. (a)
x0+
lim f (x) = -, lim f (x) = 0
x+ -1/3
y 1
(e (
e 3, 3 e
15/4 , 15 4e5/4
)
ln x (b) y = x 3 - ln x y = 3x4/3 y = 0 when x = e3 ; 4 ln x - 15 y = 9x7/3 y = 0 when x = e15/4
)
0.5
x 8 16 24 32 40
Exercise Set 5.3
183
57. (a)
0.4
0.5
3
(b) y = (1 - bx)e-bx , y = b2 (x - 2/b)e-bx ; relative maximum at x = 1/b, y = 1/be; point of inflection at x = 2/b, y = 2/be2 . Increasing b moves the relative maximum and the point of inflection to the left and down, i.e. towards the origin.
0.2
58. (a)
1
2 0
2
(b) y = -2bxe-bx , 2 y = 2b(-1 + 2bx2 )e-bx ; relative maximum at x = 0, y = 1; points of inflection at x = 1/2b, y = 1/ e. Increasing b moves the points of inflection towards the y-axis; the relative maximum doesn't move. (b)
y 6 (0,1) 4
2
59. (a) The oscillations of ex cos x about zero increase as x + so the limit does not exist, and lim ex cos x = 0.
x-
2 1
2
(1.52, 0.22) x 1 2
(c) The curve y = eax cos bx oscillates between y = eax and y = -eax . The frequency of oscillation increases when b increases.
y b=3 5 x 1 5 a=1 b=2 2 b=1 1 b=1 10 5 a=2 a=1 0.5 1 y a=3
x
60. (a)
10
(b)
y = y =
n2 - 2x2 n-1 -x2 /n x e , n
0 0
6
n4 - n3 - 4x2 n2 - 2x2 n + 4x4 n-2 -x2 /n x e , n2 3 n relative maximum when x = , y = ; e 2 point of inflection when 2n + 1 8n + 1 n. x= 4 As n increases the maxima move up and to the right; the points of inflection move to the right.
61. (a) (b) (c) (d)
x = 1, 2.5, 4 and x = 3, the latter being a cusp (-, 1], [2.5, 3) relative maxima for x = 1, 3; relative minima for x = 2.5 x = 0.8, 1.9, 4
184
Chapter 5
62. (a) f (x) = -2h(x) + (1 - 2x)h (x), f (5) = -2h(5) - 9h (5). But from the graph h (5) -0.2 and f (5) = 0, so h(5) = -(9/2)h (5) 0.9 (b) f (x) = -4h (x) + (1 - 2x)h (x), f (5) 0.8 - 9h (5) and since h (5) is clearly negative, f (5) > 0 and thus f has a minimum at x = 5. 63. Let y be the length of the other side of the rectangle, then L = 2x+2y and xy = 400 so y = 400/x and hence L = 2x + 800/x. L = 2x is an oblique asymptote L = 2x + 2(x2 + 400) 800 = , x x
L
800 2(x2 - 400) L =2- 2 = , x x2 1600 L = 3 , x L = 0 when x = 20, L = 80 64. Let y be the height of the box, then S = x2 + 4xy and x2 y = 500 so y = 500/x2 and hence S = x2 + 2000/x. The graph approaches the curve S = x2 asymptotically x3 + 2000 2000 S = x2 + = , x x S = 2x - S =2+ 2000 2(x3 - 1000) = , x2 x2
3
100 x 20
S
1000
x 30
4000 2(x + 2000) = , 3 x x3 S = 0 when x = 10, S = 300
y
65. y = 0.1x4 (6x - 5); critical numbers: x = 0, x = 5/6; relative minimum at x = 5/6, y -6.7 10-3
0.01 1
x 1
66. y = 0.1x4 (x + 1)(7x + 5); critical numbers: x = 0, x = -1, x = -5/7, relative maximum at x = -1, y = 0; relative minimum at x = -5/7, y -1.5 10-3
y
0.001 1
x
Exercise Set 5.4
185
EXERCISE SET 5.4
1. relative maxima at x = 2, 6; absolute maximum at x = 6; relative and absolute minima at x = 0, 4 2. relative maximum at x = 3; absolute maximum at x = 7; relative minima at x = 1, 5; absolute minima at x = 1, 5 3. (a)
y
(b)
y x 2 7
x 10
(c)
y x 3 5 7
4. (a)
y
(b)
y
x
x
(c)
y
x 5 5
5. x = 1 is a point of discontinuity of f . 6. Since f is monotonically increasing on (0, 1), one might expect a minimum at x = 0 and a maximum at x = 1. But both points are discontinuities of f . 7. f (x) = 8x - 12, f (x) = 0 when x = 3/2; f (1) = 2, f (3/2) = 1, f (2) = 2 so the maximum value is 2 at x = 1, 2 and the minimum value is 1 at x = 3/2. 8. f (x) = 8 - 2x, f (x) = 0 when x = 4; f (0) = 0, f (4) = 16, f (6) = 12 so the maximum value is 16 at x = 4 and the minimum value is 0 at x = 0. 9. f (x) = 3(x - 2)2 , f (x) = 0 when x = 2; f (1) = -1, f (2) = 0, f (4) = 8 so the minimum is -1 at x = 1 and the maximum is 8 at x = 4.
186
Chapter 5
10. f (x) = 6x2 + 6x - 12, f (x) = 0 when x = -2, 1; f (-3) = 9, f (-2) = 20, f (1) = -7, f (2) = 4, so the minimum is -7 at x = 1 and the maximum is 20 at x = -2. 11. f (x) = 3/(4x2 + 1)3/2 , no critical points; f (-1) = -3/ 5, f (1) = 3/ 5 so the maximum value is 3/ 5 at x = 1 and the minimum value is -3/ 5 at x = -1. 12. f (x) = 2(2x + 1) , f (x) = 0 when x = -1/2 and f (x) does not exist when x = -1, 0; 3(x2 + x)1/3 f (-2) = 22/3 , f (-1) = 0, f (-1/2) = 4-2/3 , f (0) = 0, f (3) = 122/3 so the maximum value is 122/3 at x = 3 and the minimum value is 0 at x = -1, 0.
13. f (x) = 1 - 2 cos x, f (x) = 0 when x = /3; then f (-/4) = -/4 + 2; f (/3) = /3 - 3; f (/2) = /2 - 2, so f has a minimum of /3 - 3 at x = /3 and a maximum of -/4 + 2 at x = -/4. 14. f (x) = cos x + sin x, f (x) = 0 for x in (0, ) when x = 3/4; f (0) = -1, f (3/4) = so the maximum value is 2 at x = 3/4 and the minimum value is -1 at x = 0. 15. f (x) = 1 + |9 - x2 | = 10 - x2 , |x| 3 , f (x) = -8 + x2 , |x| > 3 -2x, |x| < 3 2x, |x| > 3
x-3-
2, f () = 1
thus f (x) = 0 when
x-3+
x = 0, f (x) does not exist for x in (-5, 1) when x = -3 because
lim f (x) =
lim f (x) (see
Theorem preceding Exercise 61, Section 3.3); f (-5) = 17, f (-3) = 1, f (0) = 10, f (1) = 9 so the maximum value is 17 at x = -5 and the minimum value is 1 at x = -3. 16. f (x) = |6 - 4x| = 6 - 4x, x 3/2 , f (x) = -6 + 4x, x > 3/2 -4, x < 3/2 , f (x) does not exist when 4, x > 3/2
x = 3/2 thus 3/2 is the only critical point in (-3, 3); f (-3) = 18, f (3/2) = 0, f (3) = 6 so the maximum value is 18 at x = -3 and the minimum value is 0 at x = 3/2. 17. f (x) = 2x - 1, f (x) = 0 when x = 1/2; f (1/2) = -9/4 and minimum of -9/4 at x = 1/2 and no maximum.
x
lim f (x) = +. Thus f has a
18. f (x) = -4(x + 1); critical point x = -1. Maximum value f (-1) = 5, no minimum. 19. f (x) = 12x2 (1 - x); critical points x = 0, 1. Maximum value f (1) = 1, no minimum because lim f (x) = -.
x+
20. f (x) = 4(x3 + 1); critical point x = -1. Minimum value f (-1) = -3, no maximum. 21. No maximum or minimum because lim f (x) = + and lim f (x) = -.
x+ x-
22. No maximum or minimum because lim f (x) = + and lim f (x) = -.
x+ x-
23.
x-1-
lim f (x) = -, so there is no absolute minimum on the interval;
f (x) = 0 at x -2.414213562, for which y -4.828427125. Also f (-5) = -13/2, so the absolute maximum of f on the interval is y -4.828427125 taken at x -2.414213562. 24.
x-1+
lim f (x) = -, so there is no absolute minimum on the interval.
f (x) = 3/(x + 1)2 > 0, so f is increasing on the interval (-1, 5] and the maximum must occur at the endpoint x = 5 where f (5) = 1/2.
Exercise Set 5.4
187
25.
f (x) = 4x(x - 2)(x - 1), f (x) = 0 when x = 0, 1, 2, and f (0) = 0, f (1) = 1, f (2) = 0 so f has an absolute minimum of 0 at x = 0, 2.
x
lim = + so there is no absolute maximum.
8
2 0
4
26. (x-1)2 (x+2)2 can never be less than zero because it is the product of two squares; the minimum value is 0 for x = 1 or -2, no maximum because lim f (x) = +.
x+
15
3 0
2
27. f (x) =
5(8 - x) , f (x) = 0 when x = 8 and f (x) 3x1/3 does not exist when x = 0; f (-1) = 21, f (0) = 0, f (8) = 48, f (20) = 0 so the maximum value is 48 at x = 8 and the minimum value is 0 at x = 0, 20.
50
1 0
20
28. f (x) = (2 - x2 )/(x2 + 2)2 , f (x) = 0 for x in the interval (-1, 4) when x = 2; f (-1) = -1/3, f ( 2) = 2/4, f (4) = 2/9 so the maximum value is 2/4 at x = 2 and the minimum value is -1/3 at x = -1.
0.4
1
4
0.4
29. f (x) = -1/x2 ; no maximum or minimum because there are no critical points in (0, +).
25
0 0
10
188
Chapter 5
30. f (x) = -
x(x - 2) , and for 1 x < +, f (x) = 0 - 2x + 2)2 when x = 2. Also lim f (x) = 2 and f (2) = 5/2 and (x2
x+
3
f (1) = 2, hence f has an absolute minimum value of 2 at x = 1 and an absolute maximum value of 5/2 at x = 2.
1 0 8
31. f (x) =
1 - 2 cos x ; f (x) = 0 on [/4, 3/4] only when sin2 x x = /3. Then f (/4) = 2 2 - 1, f (/3) = 3 and f (3/4) = 2 2 + 1, so f has an absolute maximum value of 2 2 + 1 at x = 3/4 and an absolute minimum value of 3 at x = /3.
3
3
9 0
32. f (x) = 2 sin x cos x-sin x = sin x(2 cos x-1), f (x) = 0 for x in (-, ) when x = 0, /3; f (-) = -1, f (-/3) = 5/4, f (0) = 1, f (/3) = 5/4, f () = -1 so the maximum value is 5/4 at x = /3 and the minimum value is -1 at x = .
1.5
C
c
1.5
33. f (x) = x2 (3 - 2x)e-2x , f (x) = 0 for x in [1, 4] when 27 x = 3/2; if x = 1, 3/2, 4, then f (x) = e-2 , e-3 , 64e-8 ; 8 27 -3 e at critical point at x = 3/2; absolute maximum of 8 -8 x = 3/2, absolute minimum of 64e at x = 4
0.2
1 0
4
34. f (x) = (1 - ln 2x)/x2 , f (x) = 0 on [1, e] for x = e/2; if x = 1, e/2, e then f (x) = ln 2, 2/e, (ln 2 + 1)/e; 1 + ln 2 at x = e, absolute minimum of e absolute maximum of 2/e at x = e/2
0.76
1 0.64
2.7
Exercise Set 5.4
189
35. f (x) = -
3x2 - 10x + 3 1 , f (x) = 0 when x = , 3. 2+1 x 3 1 10 Then f (0) = 0, f = 5 ln - 1, 3 9
0
3.0
f (3) = 5 ln 10 - 9, f (4) = 5 ln 17 - 12 and thus f has an absolute minimum of 5(ln 10 - ln 9) - 1 at x = 1/3 and an absolute maximum of 5 ln 10 - 9 at x = 3. 36. f (x) = (x2 + 2x - 1)ex , f (x) = 0 at x = -2 + 2 and (-1+2) x = -1 - 2 (discard), f (-1 + 2) = (2 - 2 2)e -1.25, absolute maximum at x 2, f (2) = 3e2 22.17, = absolute minimum at x = -1 + 2
4
2.5
20
2
2
20
37. f (x) = -[cos(cos x)] sin x; f (x) = 0 if sin x = 0 or if cos(cos x) = 0. If sin x = 0, then x = is the critical point in (0, 2); cos(cos x) = 0 has no solutions because -1 cos x 1. Thus f (0) = sin(1), f () = sin(-1) = - sin(1), and f (2) = sin(1) so the maximum value is sin(1) 0.84147 and the minimum value is - sin(1) -0.84147.
1
0
o
1
38. f (x) = -[sin(sin x)] cos x; f (x) = 0 if cos x = 0 or if sin(sin x) = 0. If cos x = 0, then x = /2 is the critical point in (0, ); sin(sin x) = 0 if sin x = 0, which gives no critical points in (0, ). Thus f (0) = 1, f (/2) = cos(1), and f () = 1 so the maximum value is 1 and the minimum value is cos(1) 0.54030.
1.5
0 0
c
39. f (x) =
4, 2x - 5,
x1
x<1 x>1
x1
so f (x) = 0 when x = 5/2, and f (x) does not exist when x = 1
because lim f (x) = lim f (x) (see Theorem preceding Exercise 61, Section 3.3); f (1/2) = 0, - + f (1) = 2, f (5/2) = -1/4, f (7/2) = 3/4 so the maximum value is 2 and the minimum value is -1/4. 40. f (x) = 2x + p which exists throughout the interval (0, 2) for all values of p so f (1) = 0 because f (1) is an extreme value, thus 2 + p = 0, p = -2. f (1) = 3 so 12 + (-2)(1) + q = 3, q = 4 thus f (x) = x2 - 2x + 4 and f (0) = 4, f (2) = 4 so f (1) is the minimum value. 41. The period of f (x) is 2, so check f (0) = 3, f (2) = 3 and the critical points. f (x) = -2 sin x - 2 sin 2x = -2 sin x(1 + 2 cos x) = 0 on [0, 2] at x = 0, , 2 and x = 2/3, 4/3. Check f () = -1, f (2/3) = -3/2, f (4/3) = -3/2. Thus f has an absolute maximum on (-, +) of 3 at x = 2k, k = 0, 1, 2, . . . and an absolute minimum of -3/2 at x = 2k 2/3, k = 0, 1, 2, . . ..
190
Chapter 5
42. cos
x x has a period of 6, and cos a period of 4, so f (x) has a period of 12. Consider the 3 2 x x x x interval [0, 12]. f (x) = - sin - sin , f (x) = 0 when sin + sin = 0 thus, by use of 3 2 3 2 a-b 5x x 5x a+b cos , 2 sin cos - = 0 so sin = 0 or the trig identity sin a + sin b = 2 sin 2 2 12 12 12 cos
5x x x = 0. Solve sin = 0 to get x = 12/5, 24/5, 36/5, 48/5 and then solve cos =0 12 12 12 to get x = 6. The corresponding values of f (x) are -4.0450, 1.5450, 1.5450, -4.0450, 1, 5, 5 so the maximum value is 5 and the minimum value is -4.0450 (approximately). 43. Let f (x) = x - sin x, then f (x) = 1 - cos x and so f (x) = 0 when cos x = 1 which has no solution for 0 < x < 2 thus the minimum value of f must occur at 0 or 2. f (0) = 0, f (2) = 2 so 0 is the minimum value on [0, 2] thus x - sin x 0, sin x x for all x in [0, 2]. 44. Let h(x) = cos x - 1 + x2 /2. Then h(0) = 0, and it is sufficient to show that h (x) 0 for 0 < x < 2. But h (x) = - sin x + x 0 by Exercise 43. 45. Let m = slope at x, then m = f (x) = 3x2 - 6x + 5, dm/dx = 6x - 6; critical point for m is x = 1, minimum value of m is f (1) = 2 46. (a) lim f (x) = +, lim f (x) = +, so f has no maximum value on the interval. By
x0+
x(/2)-
table 5.4.3 f must have a minimum value. (b) According to table 5.4.3, there is an absolute minimum value of f on (0, /2). To find the absolute minimum value, we examine the critical points (Theorem 5.4.3). f (x) = sec x tan x - csc x cot x = 0 at x = /4, where f (/4) = 2 2, which must be the absolute minimum value of f on the interval (0, /2). 47.
x+
lim f (x) = +, lim f (x) = +, so there is no absolute maximum value of f for x > 8. By
x8-
2x(-520 + 192x - 24x2 + x3 ) , we must solve (x - 8)3 a quartic equation to find the critical points. But it is easy to see that x = 0 and x = 10 are real roots, and the other two are complex. Since x = 0 is not in the interval in question, we must have an absolute minimum of f on (8, +) of 125 at x = 10. table 5.4.3 there must be a minimum. Since f (x) = 48. (a) K ln(a/b) dC dC = ae-at - be-bt so = 0 at t = . This is the only stationary point and dt a-b dt a-b C(0) = 0, lim C(t) = 0, C(t) > 0 for 0 < t < +, so it is an absolute maximum.
t+
(b)
0.7
0 0
10
49. The slope of the line is -1, and the slope of the tangent to y = -x2 is -2x so -2x = -1, x = 1/2. The line lies above the curve so the vertical distance is given by F (x) = 2 - x + x2 ; F (-1) = 4, F (1/2) = 7/4, F (3/2) = 11/4. The point (1/2, -1/4) is closest, the point (-1, -1) farthest.
Exercise Set 5.5
191
50. The slope of the line is 4/3; and the slope of the tangent to y = x3 is 3x2 so 3x2 = 4/3, x2 = 4/9, x = 2/3. The line lies below the curve so the vertical distance is given by F (x) = x3 - 4x/3 + 1; F (-1) = 4/3, F (-2/3) = 43/27, F (2/3) = 11/27, F (1) = 2/3. The closest point is (2/3, 8/27), the farthest is (-2/3, -8/27). 51. The absolute extrema of y(t) can occur at the endpoints t = 0, 12 or when dy/dt = 2 sin t = 0, i.e. t = 0, 12, k, k = 1, 2, 3; the absolute maximum is y = 4 at t = , 3; the absolute minimum is y = 0 at t = 0, 2. 52. (a) The absolute extrema of y(t) can occur at the endpoints t = 0, 2 or when dy/dt = 2 cos 2t - 4 sin t cos t = 2 cos 2t - 2 sin 2t = 0, t = 0, 2, /8, 5/8, 9/8, 13/8; the absolute maximum is y = 3.4142 at t = /8, 9/8; the absolute minimum is y = 0.5859 at t = 5/8, 13/8. (b) The absolute extrema of x(t) occur at the endpoints t = 0, 2 or when dx 2 sin t + 1 = 0, t = 7/6, 11/6. The absolute maximum is x = 0.5774 at t = 11/6 =- dt (2 + sin t)2 and the absolute minimum is x = -0.5774 at t = 7/6. 53. f (x) = 2ax + b; critical point is x = - f (x) = 2a > 0 so f f f - - b 2a b 2a =a - 0, b 2a -
2
b 2a
b 2a
is the minimum value of f , but b 2a +c= -b2 + 4ac thus f (x) 0 if and only if 4a
+b -
-b2 + 4ac 0, -b2 + 4ac 0, b2 - 4ac 0 4a
54. Use the proof given in the text, replacing "maximum" by "minimum" and "largest" by "smallest" and reversing the order of all inequality symbols.
EXERCISE SET 5.5
1. If y = x + 1/x for 1/2 x 3/2 then dy/dx = 1 - 1/x2 = (x2 - 1)/x2 , dy/dx = 0 when x = 1. If x = 1/2, 1, 3/2 then y = 5/2, 2, 13/6 so (a) y is as small as possible when x = 1. (b) y is as large as possible when x = 1/2. 2. Let x and y be nonnegative numbers and z the sum of their squares, then z = x2 + y 2 . But x + y = 1, y = 1 - x so z = x2 + (1 - x)2 = 2x2 - 2x + 1 for 0 x 1. dz/dx = 4x - 2, dz/dx = 0 when x = 1/2. If x = 0, 1/2, 1 then z = 1, 1/2, 1 so (a) z is as large as possible when one number is 0 and the other is 1. (b) z is as small as possible when both numbers are 1/2. 3. A = xy where x + 2y = 1000 so y = 500 - x/2 and A = 500x - x2 /2 for x in [0, 1000]; dA/dx = 500 - x, dA/dx = 0 when x = 500. If x = 0 or 1000 then A = 0, if x = 500 then A = 125, 000 so the area is maximum when x = 500 ft and y = 500 - 500/2 = 250 ft.
Stream
y x
192
Chapter 5
4. The triangle in the figure is determined by the two sides of length a and b and the angle . Suppose that a + b = 1000. Then h = a sin and area = Q = hb/2 = ab(sin )/2. This expression is maximized by choosing = /2 and thus Q = ab/2 = a(1000 - a)/2, which is maximized by a = 500. Thus the maximal area is obtained by a right isosceles triangle, where the angle between two sides of 500 is the right angle. 5. Let x and y be the dimensions shown in the figure and A the area, then A = xy subject to the cost condition 3(2x) + 2(2y) = 6000, or y = 1500 - 3x/2. Thus A = x(1500 - 3x/2) = 1500x - 3x2 /2 for x in [0, 1000]. dA/dx = 1500 - 3x, dA/dx = 0 when x = 500. If x = 0 or 1000 then A = 0, if x = 500 then A = 375, 000 so the area is greatest when x = 500 ft and (from y = 1500 - 3x/2) when y = 750 ft. 6. Let x and y be the dimensions shown in the figure and A the area of the rectangle, then A = xy and, by similar triangles, x/6 = (8 - y)/8, y = 8 - 4x/3 so A = x(8 - 4x/3) = 8x - 4x2 /3 for x in [0, 6]. dA/dx = 8 - 8x/3, dA/dx = 0 when x = 3. If x = 0, 3, 6 then A = 0, 12, 0 so the area is greatest when x = 3 in and (from y = 8 - 4x/3) y = 4 in.
C
c
h
a
A
b
B
Heavy-duty
Standard
y
x
10
x y 6
8
7. Let x, y, and z be as shown in the figure and A the area of the rectangle, then A = xy and, by similar triangles, z/10 = y/6, z = 5y/3; also x/10 = (8 - z)/8 = (8 - 5y/3)/8 thus y = 24/5 - 12x/25 so A = x(24/5 - 12x/25) = 24x/5 - 12x2 /25 for x in [0, 10]. dA/dx = 24/5 - 24x/25, dA/dx = 0 when x = 5. If x = 0, 5, 10 then A = 0, 12, 0 so the area is greatest when x = 5 in. and y = 12/5 in.
y 10 x 6
z 8
8. A = (2x)y = 2xy where y = 16 - x2 so A = 32x - 2x3 for 0 x 4; dA/dx = 32 - 6x2 , dA/dx = 0 when x = 4/ 3. If x = 0, 4/ 3, then A = 0, 256/(3 3), 0 so the area is largest 4 when x = 4/ 3 and y = 32/3. The dimensions of the rectangle with largest area are 8/ 3 by 32/3.
y 16
y x 4 x 4
9. A = xy where x2 + y 2 = 202 = 400 so y = 400 - x2 and A = x 400 - x2 for 0 x 20; 2, dA/dx = 2(200 x2 )/ 400 - x dA/dx = 0 when - A the x = 200 = 10 2. If x = 0, 10 2, 20 then = 0, 200, 0 so area is maximum when x = 10 2 and y = 400 - 200 = 10 2.
10 x
y
Exercise Set 5.5
193
10. Let R denote the rectangle with corners (8, 10). Suppose the lower left corner of the square S is at the point (x0 , y0 ) = (x0 , -4x0 ). Let Q denote the desired region.
y = 4x
S
It is clear that Q will be a rectangle, and that its left and bottom sides are subsets of the left and bottom sides of S. The right and top edges of S will be subsets of the sides of R (if S is big enough). From the drawing it is evident that the area of Q is (8-x0 )(10+ 4x0 ) = -4x2 + 22x0 + 80. This function is maximized when 0 x0 = 11/4, for which the area of Q is 441/4. The maximum possible area for Q is 441/4, taken when x0 = 11/4.
(x 0, y 0)
6 2
R
6
y = 4x
11. Let x = length of each side that uses the $1 per foot fencing, y = length of each side that uses the $2 per foot fencing. The cost is C = (1)(2x) + (2)(2y) = 2x + 4y, but A = xy = 3200 thus y = 3200/x so C = 2x + 12800/x for x > 0, dC/dx = 2 - 12800/x2 , dC/dx = 0 when x = 80, d2 C/dx2 > 0 so C is least when x = 80, y = 40. 12. A = xy where 2x + 2y = p so y = p/2 - x and A = px/2 - x2 for x in [0, p/2]; dA/dx = p/2 - 2x, dA/dx = 0 when x = p/4. If x = 0 or p/2 then A = 0, if x = p/4 then A = p2 /16 so the area is maximum when x = p/4 and y = p/2 - p/4 = p/4, which is a square.
y x
13. Let x and y be the dimensions of a rectangle; the perimeter is p = 2x + 2y. But A = xy thus y = A/x so p = 2x + 2A/x for x > 0, dp/dx = 2 - 2A/x2 = 2(x2 - A)/x2 , dp/dx = 0 when x = A, d2 p/dx2 = 4A/x3 > 0 if x > 0 so p is a minimum when x = A and y = A and thus the rectangle is a square. 14. With x, y, r, and s as shown in the figure, the sum of the y x and s = because enclosed areas is A = r2 + s2 where r = 2 4 x is the circumference of the circle and y is the perimeter of the x2 y2 square, thus A = + . But x + y = 12, so y = 12 - x and 4 16 A= (12 - x) +4 2 3 x + = x - x + 9 for 0 x 12. 4 16 16 2
2 2
x cut
12
y
r
s
dA 3 dA 12 12 +4 x- , = 0 when x = . If x = 0, , 12 = dx 8 2 dx +4 +4 36 36 then A = 9, , so the sum of the enclosed areas is +4 (a) a maximum when x = 12 in. (when all of the wire is used for the circle) (b) a minimum when x = 12/( + 4) in.
194
Chapter 5
15. (a)
dN = 250(20 - t)e-t/20 = 0 at t = 20, N (0) = 125,000, N (20) 161,788, and dt N (100) 128,369; the absolute maximum is N = 161788 at t = 20, the absolute minimum is N = 125,000 at t = 0. d2 N dN occurs when (b) The absolute minimum of = 12.5(t - 40)e-t/20 = 0, t = 40. dt dt2
16. The area of the window is A = 2rh + r2 /2, the perimeter 1 is p = 2r + 2h + r thus h = [p - (2 + )r] so 2 A = r[p - (2 + )r] + r2 /2 = pr - (2 + /2)r2 for 0 r p/(2 + ), dA/dr = p - (4 + )r, dA/dr = 0 when r = p/(4 + ) and d2 A/dr2 < 0, so A is maximum when r = p/(4 + ).
2r
r
h
17. Let the box have dimensions x, x, y, with y x. The constraint is 4x + y 108, and the volume V = x2 y. If we take y = 108 - 4x then V = x2 (108 - 4x) and dV /dx = 12x(-x + 18) with roots x = 0, 18. The maximum value of V occurs at x = 18, y = 36 with V = 11, 664 in2 . 18. Let the box have dimensions x, x, y with x y. The constraint is x+2(x+y) 108, and the volume V = x2 y. Take x = (108 - 2y)/3 = 36 - 2y/3, V = y(36 - 2y/3)2 , dV /dy = (4/3)y 2 - 96y + 1296 with roots y = 18, 54. Then d2 V /dy 2 = (8/3)y - 96 is negative for y = 18, so by the second derivative test, V has a maximum of 10, 368 in2 at y = 18, x = 24. 19. Let x be the length of each side of a square, then V = x(3 - 2x)(8 - 2x) = 4x3 - 22x2 + 24x for 0 x 3/2; dV /dx = 12x2 - 44x + 24 = 4(3x - 2)(x - 3), dV /dx = 0 when x = 2/3 for 0 < x < 3/2. If x = 0, 2/3, 3/2 then V = 0, 200/27, 0 so the maximum volume is 200/27 ft3 . 20. Let x = length of each edge of base, y = height. The cost is C = (cost of top and bottom) + (cost of sides) = (2)(2x2 ) + (3)(4xy) = 4x2 + 12xy, but V = x2 y = 2250 thus y = 2250/x2 so C = 4x2 + 27000/x for x > 0, dC/dx = 8x - 27000/x2 , dC/dx = 0 when x = 3 3375 = 15, d2 C/dx2 > 0 so C is least when x = 15, y = 10. 21. Let x = length of each edge of base, y = height, k = $/cm2 for the sides. The cost is C = (2k)(2x2 ) + (k)(4xy) = 4k(x2 + xy), but V = x2 y = 2000 thus y = 2000/x2 so C = 4k(x2 + 2000/x) for x > 0, dC/dx = 4k(2x - 2000/x2 ), dC/dx = 0 when x = 3 1000 = 10, d2 C/dx2 > 0 so C is least when x = 10, y = 20. 22. Let x and y be the dimensions shown in the figure and V the volume, then V = x2 y. The amount of material is to be 1000 ft2 , thus (area of base) + (area of sides) = 1000, x2 + 4xy = 1000, 1000 - x2 1000 - x2 1 y = so V = x2 = (1000x - x3 ) for 4x 4x 4 1 dV dV = (1000 - 3x2 ), =0 0 < x 10 10. dx 4 dx 1000/3 = 10 10/3. 5000 If x = 0, 10 10/3, 10 10 then V = 0, 3 when x =
y
x x
10/3, 0;
the volume is greatest for x = 10 10/3 ft and y = 5 10/3 ft.
Exercise Set 5.5
195
23. Let x = height and width, y = length. The surface area is S = 2x2 + 3xy where x2 y = V , so y = V /x2 and S = 2x2 + 3V /x for x > 0; dS/dx = 4x - 3V /x2 , dS/dx = 0 when x = 3 3V /4, 4 3 3V 3 3V ,y= . d2 S/dx2 > 0 so S is minimum when x = 4 3 4 24. Let r and h be the dimensions shown in the figure, then the volume of the inscribed cylinder is V = r2 h. But r2 + h 2
2
= R2 thus r2 = R2 - h2 4
h2 4 h3 4
h h 2 R
so V = R2 - for 0 h 2R.
h = R2 h -
dV dV 3 = R 2 - h2 , =0 dh 4 dh when h = 2R/ 3. If h = 0, 2R/ 3, 2R 4 then V = 0, R3 , 0 so the volume is largest 3 3 when h = 2R/ 3 and r = 2/3R. 25. Let r and h be the dimensions shown in the figure, then the surface area is S = 2rh + 2r2 . 2 h But r2 + = R2 thus h = 2 R2 - r2 so 2 S = 4r R2 - r2 + 2r2 for 0 r R, dS 4(R2 - 2r2 ) dS = = 0 when + 4r; 2 - r2 dr dr R R2 - 2r2 = -r R2 - r 2 R2 - 2r2 = -r R2 - r2 R4 - 4R2 r2 + 4r4 = r2 (R2 - r2 ) 5r4 - 5R2 r2 + R4 = 0 and using the quadratic formula r2 = (1)
r
h h 2 R
r
25R4 - 20R4 5 5 5 5 2 = R , r = R, of 10 10 10 5+ 5 5+ 5 R satisfies (1). If r = 0, R, 0 then S = 0, (5 + 5)R2 , 2R2 so which only r = 10 10 5+ 5 5- 5 2 - r2 , h = 2 the surface area is greatest when r = R and, from h = 2 R R. 10 10 5R2
196
Chapter 5
26. Let R and H be the radius and height of the cone, and r and h the radius and height of the cylinder (see figure), then the volume of the cylinder is V = r2 h. r H -h = thus By similar triangles (see figure) H R H H H h = (R - r) so V = (R - r)r2 = (Rr2 - r3 ) R R R H dV H 2 = (2Rr-3r ) = r(2R-3r), for 0 r R. dr R R dV = 0 for 0 < r < R when r = 2R/3. If dr r = 0, 2R/3, R then V = 0, 4R2 H/27, 0 so the maximum volume is cone). 4 1 2 4R2 H 4 = R H = (volume of 27 9 3 9
r H h
R
27. From (13), S = 2r2 + 2rh. But V = r2 h thus h = V /(r2 ) and so S = 2r2 + 2V /r for r > 0. dS/dr = 4r - 2V /r2 , dS/dr = 0 if r = 3 V /(2). Since d2 S/dr2 = 4 + 4V /r3 > 0, the minimum surface area is achieved when r = 3 V /2 and so h = V /(r2 ) = [V /(r3 )]r = 2r. 28. V = r2 h where S = 2r2 + 2rh so h = 1 S - 2r2 , V = (Sr - 2r3 ) for r > 0. 2r 2 1 d2 V dV = (S - 6r2 ) = 0 if r = S/(6), = -6r < 0 so V is maximum when dr 2 dr2 S - 2r2 S - 2r2 S - S/3 = r = 2r, thus the height is equal to the r= r = S/(6) and h = 2r 2r2 S/3 diameter of the base.
29. The surface area is S = r2 + 2rh where V = r2 h = 500 so h = 500/(r2 ) and S = r2 + 1000/r for r > 0; dS/dr = 2r - 1000/r2 (2r3 = - 1000)/r2 , dS/dr = 0 when r = 3 500/, d2 S/dr2 > 0 for r > 0 so S is minimum when r = 500 500 2/3 h= 2 = r 500 3 = 500/ cm
3
r
h
500/ cm and
30. The total area of material used is A = Atop + Abottom + Aside = (2r)2 + (2r)2 + 2rh = 8r2 + 2rh. The volume is V = r2 h thus h = V /(r2 ) so A = 8r2 + 2V /r for r > 0, dA/dr = 16r - 2V /r2 = 2(8r3 - V )/r2 , dA/dr = 0 when r = 3 V /2. This is the only critical point, r r = r3 and, for d2 A/dr2 > 0 there so the least material is used when r = 3 V /2, = 2) h V /(r V r V 3 r = V /2, = = . h V 8 8 31. Let x be the length of each side of the squares and y the height of the frame, then the volume is V = x2 y. The total length of the wire is L thus 8x + 4y = L, y = (L - 8x)/4 so V = x2 (L - 8x)/4 = (Lx2 - 8x3 )/4 for 0 x L/8. dV /dx = (2Lx - 24x2 )/4, dV /dx = 0 for 0 < x < L/8 when x = L/12. If x = 0, L/12, L/8 then V = 0, L3 /1728, 0 so the volume is greatest when x = L/12 and y = L/12.
Exercise Set 5.5
197
32. (a) Let x = diameter of the sphere, y = length of an edge of the cube. The combined volume is 1 (S - x2 )1/2 V = x3 + y 3 and the surface area is S = x2 + 6y 2 = constant. Thus y = 6 61/2 S 3 (S - x2 )3/2 ; and V = x + for 0 x 6 63/2 dV 3 dV = x2 - 3/2 x(S - x2 )1/2 = x( 6x - S - x2 ). = 0 when x = 0, or when dx 2 dx 6 2 6 S S S S , x = . If x = 0, , , 6x = S - x2 , 6x2 = S - x2 , x2 = 6+ 6+ 6+ 3/2 3/2 3/2 S S S S then V = 3/2 , , and hence when , so that V is smallest when x = 6+ 6 6 6+ 6 S y= , thus x = y. 6+ (b) From Part (a), the sum of the volumes is greatest when there is no cube. 33. Let h and r be the dimensions shown in the figure, then the 1 volume is V = r2 h. But r2 + h2 = L2 thus r2 = L2 - h2 3 1 1 dV 2 = so V = (L -h2 )h = (L2 h-h3 ) for 0 h L. 3 3 dh 1 dV (L2 - 3h2 ). = 0 when h = L/ 3. If h = 0, L/ 3, 0 3 dh 2 then V = 0, L3 , 0 so the volume is as large as possible 9 3 when h = L/ 3 and r = 2/3L. 34. Let r and h be the radius and height of the cone (see figure). The slant height of any such cone will be R, the radius of the circular sheet. Refer to the solution of Exercise 33 to 2 find that the largest volume is R3 . 9 3
h
L
r
h
R
r
35. The area of the paper is A = rL = r r2 + h2 , but 1 V = r2 h = 10 thus h = 30/(r2 ) 3 so A = r r2 + 900/( 2 r4 ).
h
r
To simplify the computations let S = A2 , S = 2 r2 r2 + 900 2 r4 = 2 r4 + 900 for r > 0, r2
L
dS 1800 4( 2 r6 - 450) = 4 2 r3 - 3 = , dS/dr = 0 when dr r r3 450/ 2 , d2 S/dr2 > 0, so S and hence A is least 30 3 2 /450 cm. when r = 6 450/ 2 cm, h = r=
6
198
Chapter 5
36. The area of the triangle is A =
1 hb. By similar trian2 R b/2 2Rh = gles (see figure) ,b= h h2 - 2Rh h2 - 2Rh so A = Rh2 (h - 3R) Rh2 dA = 2 , for h > 2R, dh (h - 2Rh)3/2 h2 - 2Rh
h
h-R
h2 - 2Rh
R R b/2 b
dA = 0 for h > 2R when h = 3R, by the first derivadh tive test A is minimum when h = 3R. If h = 3R then b = 2 3R (the triangle is equilateral). 37. The volume of the cone is V = 1 2 r h. By similar tri3
r Rh R angles (see figure) ,r= = 2 - 2Rh 2 - 2Rh h h h 1 h3 h2 1 = R2 for h > 2R, so V = R2 2 3 h - 2Rh 3 h - 2R dV h(h - 4R) dV 1 , = R2 = 0 for h > 2R when dh 3 (h - 2R)2 dh h = 4R, by the first derivative test V is minimum when h = 4R. If h = 4R then r = 2R. 38. The area is (see figure) 1 A = (2 sin )(4 + 4 cos ) 2 = 4(sin + sin cos ) for 0 /2; dA/d = 4(cos - sin2 + cos2 ) = 4(cos - [1 - cos2 ] + cos2 ) = 4(2 cos2 + cos - 1) = 4(2 cos - 1)(cos + 1) dA/d = 0 when = /3 for 0 < < /2. If = 0, /3, /2 then A = 0, 3 3, 4 so the maximum area is 3 3. 39. Let b and h be the dimensions shown in the figure, 1 then the cross-sectional area is A = h(5 + b). But 2 h = 5 sin and b = 5 + 2(5 cos ) = 5 + 10 cos so 5 sin (10 + 10 cos ) = 25 sin (1 + cos ) for A = 2 0 /2. dA/d = -25 sin2 + 25 cos (1 + cos ) = 25(- sin2 + cos + cos2 ) = 25(-1 + cos2 + cos + cos2 ) = 25(2 cos2 + cos - 1) = 25(2 cos - 1)(cos + 1). dA/d = 0 for 0 < < /2 when = 1/2, = /3. cos If = 0, /3, /2 then A = 0, 75 3/4, 25 so the crosssectional area is greatest when = /3.
h-R
h2 - 2Rh
h R R
r
4 cos 2
2 sin
4
2 cos
5 cos
b
5
h = 5 sin
5
5
Exercise Set 5.5
199
40. I = k
, k the constant of proportionality. If h is the height of the lamp above the table then r2 - 2h2 h h dI dI =k 2 =0 for h > 0, , cos = h/ and = h2 + r2 so I = k 3 = k 2 dh (h + r2 )3/2 (h + r2 )5/2 dh when h = r/ 2, by the first derivative test I is maximum when h = r/ 2.
2
cos
41. Let L, L1 , and L2 be as shown in the figure, then L = L1 + L2 = 8 csc + sec , dL = -8 csc cot + sec tan , 0 < < /2 d -8 cos3 + sin3 sin 8 cos = ; + =- 2 cos2 sin sin2 cos2
L2 L L1 8
dL = 0 if sin3 = 8 cos3 , tan3 = 8, tan = 2 which gives d the absolute minimum for L because lim L = lim L = +. 0+ /2- If tan = 2, then csc = 5/2 and sec = 5 so L = 8( 5/2) + 5 = 5 5 ft. 42. Let
1
x = number of steers per acre w = average market weight per steer T = total market weight per acre then T = xw where w = 2000 - 50(x - 20) = 3000 - 50x so T = x(3000 - 50x) = 3000x - 50x2 for 0 x 60, dT /dx = 3000 - 100x and dT /dx = 0 when x = 30. If x = 0, 30, 60 then T = 0, 45000, 0 so the total market weight per acre is largest when 30 steers per acre are allowed.
43. (a) The daily profit is P = (revenue) - (production cost) = 100x - (100, 000 + 50x + 0.0025x2 ) = -100, 000 + 50x - 0.0025x2 for 0 x 7000, so dP/dx = 50 - 0.005x and dP/dx = 0 when x = 10, 000. Because 10,000 is not in the interval [0, 7000], the maximum profit must occur at an endpoint. When x = 0, P = -100, 000; when x = 7000, P = 127, 500 so 7000 units should be manufactured and sold daily. (b) Yes, because dP/dx > 0 when x = 7000 so profit is increasing at this production level. (c) dP/dx = 15 when x = 7000, so P (7001) - P (7000) 15, and the marginal profit is $15. 44. (a) R(x) = px but p = 1000 - x so R(x) = (1000 - x)x (b) P (x) = R(x) - C(x) = (1000 - x)x - (3000 + 20x) = -3000 + 980x - x2 (c) P (x) = 980 - 2x, P (x) = 0 for 0 < x < 500 when x = 490; test the points 0, 490, 500 to find that the profit is a maximum when x = 490. (d) P (490) = 237,100 (e) p = 1000 - x = 1000 - 490 = 510. 45. The profit is P = (profit on nondefective) - (loss on defective) = 100(x - y) - 20y = 100x - 120y but y = 0.01x + 0.00003x2 so P = 100x - 120(0.01x + 0.00003x2 ) = 98.8x - 0.0036x2 for x > 0, dP/dx = 98.8 - 0.0072x, dP/dx = 0 when x = 98.8/0.0072 13, 722, d2 P/dx2 < 0 so the profit is maximum at a production level of about 13,722 pounds.
200
Chapter 5
46. To cover 1 mile requires 1/v hours, and 1/(10 - 0.07v) gallons of diesel fuel, so the total cost to 1.50 dC 0.0315v 2 + 21v - 1500 15 + , = the client is C = . By the second derivative test, C v 10 - 0.07v dv v 2 (0.07v - 10)2 has a minimum of 50.6 cents/mile at v = 65.08 miles per hour. 47. The distance between the particles is D = (1 - t - t)2 + (t - 2t)2 = 5t2 - 4t + 1 for t 0. For convenience, we minimize D2 instead, so D2 = 5t2 - 4t + 1, dD2 /dt = 10t - 4, which is 0 when t = 2/5. d2 D2 /dt2 > 0 so D2 and hence D is minimum when t = 2/5. The minimum distance is D = 1/ 5. 48. The distance between the particles is D = (2t - t)2 + (2 - t2 )2 = t4 - 3t2 + 4 for t 0. For convenience we minimize D2 instead so D2 = t4 - 3t2 + 4, dD2 /dt = 4t3 - 6t = 4t(t2 - 3/2), which is 0 for t > 0 when t = 3/2. d2 D2 /dt2 = 12t2 - 6 > 0 when t = 3/2 so D2 and hence D is minimum there. The minimum distance is D = 7/2. 49. Let P (x, y) be a point on the curve x2 + y 2 = 1. The distance between P (x, y) and P0 (2, 0) is D = (x - 2)2 + y 2 , but y 2 = 1 - x2 so D = (x - 2)2 + 1 - x2 = 5 - 4x for -1 x 1, dD 2 which has no critical points for -1 < x < 1. If x = -1, 1 then D = 3, 1 so the = - dx 5 - 4x closest point occurs when x = 1 and y = 0. 50. Let P (x, y) be a point on y = x, then the distance D between P and (2, 0) is D = (x - 2)2 + y 2 = (x - 2)2 + x = x2 - 3x + 4, for 0 x 3. For convenience we find the = extrema for D2 instead, so D2 x2 - 3x + 4, dD2 /dx = 2x - 3 = 0 when x = 3/2. If x = 0, 3/2, 3 then D2 = 4, 7/4, 4 so D = 2, 7/2, 2. The points (0, 0) and (3, 3) are at the greatest distance, and (3/2, 3/2) the shortest distance from (2, 0). 51. (a) Draw the line perpendicular to the given line that passes through Q. (b) Let Q : (x0 , y0 ). If (x, mx + b) is the point on the graph closest to Q, then the distance squared d2 = D = (x - x0 )2 + (mx + b - y0 )2 is minimized. Then dD/dx = 2(x - x0 ) + 2m(mx + b - y0 ) = 0 at the point (x, mx + b) which minimizes the distance. On the other hand, the line connecting the point Q with the line y = mx+b is perpendicular to this line provided the connecting line has slope -1/m. But this condition is (y-y0 )/(x-x0 ) = -1/m, or m(y - y0 ) = -(x - x0 ). Since y = mx + b the condition becomes m(mx + b - y0 ) = -(x - x0 ), which is the equivalent to the expression above which minimizes the distance. 52. (a) Draw the line from the point Q through the center P of the circle. The nearer and farther points are the points on C that are, respectively closest to and farthest from Q. (b) Let Q : (xQ , yQ ) and P : (xP , yP ), and let the equation of the circle be (x - xP )2 + (y - yP )2 = c2 where c is constant. Let T be the square of the distance from (x, y) to Q: T = (x - xQ )2 + (y - yQ )2 . It is desired to find the minimum (and maximum) value of T when (x, y) lies on the circle. This occurs when the derivative of T is zero, i.e. 2(x - xQ ) + 2(y - yQ )(dy/dx) = 0. Note that dy/dx here expresses the slope of the line tangent to the circle at the point (x, y). An equivalent expression is dy/dx = -(x - xQ )/(y - yQ ), which is the negative reciprocal of the slope of the line connecting (x, y) with Q. Thus the minimum and maximum distances are achieved on this line.
Exercise Set 5.5
201
53. (a) The line through (x, y) and the origin has slope y/x, and the negative reciprocal is -x/y. If (x, y) is a point on the ellipse, then x2 - xy + y 2 = 4 and, differentiating, 2x - x(dy/dx) - y + 2y(dy/dx) = 0. For the desired points we have dy/dx = -x/y, and inserting that into the previous equation results in 2x + x2 /y - y - 2x = 0, 2xy + x2 - y 2 - 2xy = 0, x2 - y 2 = 0, y = x. Inserting this into the equation of the ellipse we have x2 x2 + x2 = 4 with solutions y = = 2 x or y -x 2/ 3. Thus there are four solutions, (2, 2), (-2, -2), (2/ 3, -2/ 3) and = = (-2/ 3, 2/ 3). (b) In general, the shortest/longest distance from a point to a curve is taken on the line connecting the point to the curve which is perpendicular to the tangent line at the point in question. 54. The tangent line to the ellipse at (x, y) has slope dy/dx, where x2 - xy + y 2 = 4. This yields 2x - y - xdy/dx + 2ydy/dx = 0, dy/dx = (2x - y)/(x - 2y). The line through (x, y) that also passes through the origin has slope y/x. Check to see if the two slopes are negative reciprocals: (2x - y)/(x - 2y) = -x/y; y(2x - y) = -x(x - 2y); x = y, so the points lie on the line y = x or on y = -x.
y
2
1
x
2 1 1 2 1 2
55. If P (x0 , y0 ) is on the curve y = 1/x2 , then y0 = 1/x2 . At P the slope of the tangent line is -2/x3 0 0 1 2 2 3 so its equation is y - 2 = - 3 (x - x0 ), or y = - 3 x + 2 . The tangent line crosses the y-axis x0 x0 x0 x0 at 3 9 3 9 , and the x-axis at x0 . The length of the segment then is L = + x2 for x0 > 0. For x2 2 x4 4 0 0 0 9 9 dL2 36 9 9(x6 - 8) 0 convenience, we minimize L2 instead, so L2 = 4 + x2 , = - 5 + x0 = , which 0 x0 4 dx0 x0 2 2x5 0 2 2 d L 2 is 0 when x6 = 8, x0 = 2. 2, y0 = 1/2. 0 2 > 0 so L and hence L is minimum when x0 = dx0
56. If P (x0 , y0 ) is on the curve y = 1 - x2 , then y0 = 1 - x2 . At P the slope of the tangent line is 0 -2x0 so its equation is y - (1 - x2 ) = -2x0 (x - x0 ), or y = -2x0 x + x2 + 1. The y-intercept is 0 0 1 2 x0 + 1 and the x-intercept is (x0 + 1/x0 ) so the area A of the triangle is 2 1 2 1 3 A = (x0 + 1)(x0 + 1/x0 ) = (x0 + 2x0 + 1/x0 ) for 0 x0 1. 4 4 1 1 2 2 dA/dx0 = (3x0 + 2 - 1/x0 ) = (3x4 + 2x2 - 1)/x2 which is 0 when x2 = -1 (reject), or when 0 0 0 0 4 4 1 2 2 3 2 x0 = 1/3 so x0 = 1/ 3. d A/dx0 = (6x0 + 2/x0 ) > 0 at x0 = 1/ 3 so a relative minimum and 4 hence the absolute minimum occurs there. 57. At each point (x, y) on the curve the slope of the tangent line is m = x, 2x dy =- for any dx (1 + x2 )2
dm 2(3x2 - 1) dm , = = 0 when x = 1/ 3, by the first derivative test the only relative dx (1 + x2 )3 dx maximum occurs at x = -1/ 3, which is the absolute maximum because lim m = 0. The x tangent line has greatest slope at the point (-1/ 3, 3/4).
202
Chapter 5
58. Let x be how far P is upstream from where the man starts (see figure), then the total time to reach T is t = (time from M to P ) + (time from P to T ) x2 + 1 1 - x = + for 0 x 1, rR rW
1 x P 1 T
where rR and rW are the rates at which he can row M and walk, respectively. x dt x2 + 1 1 - x dt 1 + , = = 0 when 5x = 3 x2 + 1, (a) t = - so 2+1 3 5 dx 5 dx 3 x 25x2 = 9(x2 + 1), x2 = 9/16, x = 3/4. If x = 0, 3/4, 1 then t = 8/15, 7/15, 2/3 so the time is a minimum when x = 3/4 mile. x2 + 1 1 1 - x dt x dt (b) t = - so + , = = 0 when x = 4/3 which is not in the 2+1 4 5 dx 5 dx 4 x interval [0, 1]. Check the endpoints to find that the time is a minimum when x = 1 (he should row directly to the town). 59. With x and y as shown in the figure, the maximum length of pipe will be the smallest value of L = x + y. By similar triangles x y 8x = ,y= so 2 - 16 2 - 16 8 x x 128 dL 8x =1- 2 , for x > 4, L=x+ 2 - 16 dx (x - 16)3/2 x dL = 0 when dx (x2 - 16)3/2 = 128 x2 - 16 = 1282/3 = 16(22/3 ) x2 = 16(1 + 22/3 ) x = 4(1 + 22/3 )1/2 ,
y 8
x
x2 16 4
d2 L/dx2 = 384x/(x2 - 16)5/2 > 0 if x > 4 so L is smallest when x = 4(1 + 22/3 )1/2 . For this value of x, L = 4(1 + 22/3 )3/2 ft. 60. s = (x1 - x)2 + (x2 - x)2 + + (xn - x)2 , ds/d = -2(x1 - x) - 2(x2 - x) - - 2(xn - x), x ds/d = 0 when x (x1 - x) + (x2 - x) + + (xn - x) = 0 (x1 + x2 + xn ) - ( + x + + x) = 0 x x (x1 + x2 + + xn ) - n = 0 1 x = (x1 + x2 + + xn ), n d2 s/d2 = 2 + 2 + + 2 = 2n > 0, so s is minimum when x = x 1 (x1 + x2 + + xn ). n
Exercise Set 5.5
203
61. Let x = distance from the weaker light source, I = the intensity at that point, and k the constant of proportionality. Then I= kS 8kS + if 0 < x < 90; x2 (90 - x)2
2kS 16kS 2kS[8x3 - (90 - x)3 ] kS(x - 30)(x2 + 2700) dI =- 3 + = = 18 , 3 3 (90 - x)3 dx x (90 - x) x x3 (x - 90)3 dI dI < 0 if x < 30, and > 0 if x > 30, so the intensity is minimum at a which is 0 when x = 30; dx dx distance of 30 cm from the weaker source. 62. If f (x0 ) is a maximum then f (x) f (x0 ) for all x in some open interval containing x0 thus f (x) f (x0 ) because x is an increasing function, so f (x0 ) is a maximum of f (x) at x0 . The proof is similar for a minimum value, simply replace by . 63. = - ( + )
-1
5-x = - cot (x - 2) - cot , 4 1 d -1/4 = + 2 dx 1 + (x - 2) 1 + (5 - x)2 /16
-1
y B(5, 4) P(x, 0) A(2, 1) x
2 5 3(x2 - 2x - 7) x2 5x =- [1 + (x - 2)2 ][16 + (5 - x)2 ] 2 4 + 28 = 1 2 2, d/dx = 0 when x = 2 only 1 + 2 2 is in [2, 5]; d/dx > 0 for x in [2, 1 + 2 2), d/dx < 0 for x in (1 + 2 2, 5], is maximum when x = 1 + 2 2.
64. = - = cot-1 (x/12) - cot-1 (x/2) d 12 2 =- + dx 144 + x2 4 + x2 10(24 - x2 ) (144 + x2 )(4 + x2 ) d/dx = 0 when x = 24 = 2 6, by the first derivative test is maximum there. =
10
2 x
65. Let v = speed of light in the medium. The total time required for the light to travel from A to P to B is 1 t = (total distance from A to P to B)/v = ( (c - x)2 + a2 + x2 + b2 ), v dt 1 - = dx v and c-x (c - x)2 + a2 + x2 x + b2 c-x (c - x)2 + a2 . But x/ x2 + b2 = sin 2 and
dt x = = 0 when dx x2 + b2
(c - x)/ (c - x)2 + a2 = sin 1 thus dt/dx = 0 when sin 2 = sin 1 so 2 = 1 .
204
Chapter 5
66. The total time required for the light to travel from A to P to B is (c - x)2 + b2 x2 + a2 + , t = (time from A to P ) + (time from P to B) = v1 v2 x dt c-x = - but x/ x2 + a2 = sin 1 and dx v1 x2 + a2 v2 (c - x)2 + b2 (c - x)/ (c - x)2 + b2 = sin 2 thus sin 1 sin 1 dt sin 2 dt sin 2 = = 0 when - so = . dx v1 v2 dx v1 v2
67. (a) The rate at which the farmer walks is analogous to the speed of light in Fermat's principle. (b) the best path occurs when 1 = 2 (see figure).
Barn
(c) by similar triangles, x/(1/4) = (1 - x)/(3/4) 3x = 1 - x 4x = 1 x = 1/4 mi.
House
1 4
2
1
3 4
x
1-x
EXERCISE SET 5.6
1. f (x) = x2 - 2, f (x) = 2x, xn+1 = xn - x2 - 2 n 2xn x1 = 1, x2 = 1.5, x3 = 1.416666667, . . . , x5 = x6 = 1.414213562 x2 - 5 n 2xn x1 = 2, x2 = 2.25, x3 = 2.236111111, x4 = 2.2360679779, x4 = x5 = 2.2360679775 x3 - 6 n 3x2 n x1 = 2, x2 = 1.833333333, x3 = 1.817263545, . . . , x5 = x6 = 1.817120593
2. f (x) = x2 - 5, f (x) = 2x, xn+1 = xn -
3. f (x) = x3 - 6, f (x) = 3x2 , xn+1 = xn -
4. xn - a = 0 5. f (x) = x3 - 2x - 2, f (x) = 3x2 - 2, xn+1 = xn - x3 - 2xn - 2 n 3x2 - 2 n
x1 = 2, x2 = 1.8, x3 = 1.7699481865, x4 = 1.7692926629, x5 = x6 = 1.7692923542 6. f (x) = x3 + x - 1, f (x) = 3x2 + 1, xn+1 = xn - x3 + xn - 1 n 3x2 + 1 n x1 = 1, x2 = 0.75, x3 = 0.686046512, . . . , x5 = x6 = 0.682327804
7. f (x) = x5 + x4 - 5, f (x) = 5x4 + 4x3 , xn+1 = xn -
x5 + x4 - 5 n n 5x4 + 4x3 n n x1 = 1, x2 = 1.333333333, x3 = 1.239420573, . . . , x6 = x7 = 1.224439550
Exercise Set 5.6
205
8. f (x) = x5 - 3x + 3, f (x) = 5x4 - 3, xn+1 = xn -
x5 - 3xn + 3 n 5x4 - 3 n
x1 = -1.5, x2 = -1.49579832, x3 = -1.49577135 = x4 = x5 9. f (x) = x4 + x2 - 4, f (x) = 4x3 + 2x, xn+1 = xn - x4 + x2 - 4 n n 4x3 + 2xn n
2.2 5 2.2 16
x1 = 1, x2 = 1.3333, x3 = 1.2561, x4 = 1.24966, . . . , x7 = x8 = 1.249621068; by symmetry, x = -1.249621068 is the other solution.
10. f (x) = x5 - 5x3 - 2, f (x) = 5x4 - 15x2 , xn+1 = xn - x5 - 5x3 - 2 n n 5x4 - 15x2 n n
2.5
10
2.5
x1 = 2, x2 = 2.5, x3 = 2.327384615, . . . , x7 = x8 = 2.273791732
20
11. f (x) = 2 cos x - x, f (x) = -2 sin x - 1 2 cos x - x xn+1 = xn - -2 sin x - 1 x1 = 1, x2 = 1.03004337, x3 = 1.02986654, x4 = x5 = 1.02986653
O
8
o
6
12. f (x) = sin x - x2 , f (x) = cos x - 2x, xn+1 sin xn - x2 n = xn - cos xn - 2xn
0.3 0 1.5
x1 = 1, x2 = 0.891395995, x3 = 0.876984845, . . . , x5 = x6 = 0.876726215
1.3
13. f (x) = x - tan x, f (x) = 1 - sec2 x = - tan2 x, xn+1 = xn + xn - tan xn tan2 xn
100
6
i
x1 = 4.5, x2 = 4.493613903, x3 = 4.493409655, x4 = x5 = 4.493409458
100
206
Chapter 5
14. f (x) = 1 + ex sin x, f (x) = ex (cos x + sin x) xn+1 = xn - 1 + e sin xn exn (cos xn + sin xn )
xn
3 0 c
x1 = 3, x2 = 3.2249, x3 = 3.1847,. . . , x10 = x11 = 3.183063012
22
15. The graphs of y = x3 and y = 1 - x intersect near the point x = 0.7. Let f (x) = x3 + x - 1, so that f (x) = 3x2 + 1, and xn+1 = xn - x3 + x - 1 . 3x2 + 1
4
1 1 2
2
If x1 = 0.7 then x2 = 0.68259109, x3 = 0.68232786, x4 = x5 = 0.68232780. 16. The graphs of y = sin x and y = x3 - 2x2 + 1 intersect at points near x = -0.8 and x = 0.6 and x = 2. Let f (x) = sin x-x3 +2x2 -1, then f (x) = cos x-3x2 +4x, so xn+1 = xn - cos x - 3x2 + 4x . sin x - x3 + 2x2 + 1
1.5
2.5
If x1 = -0.8, then x2 = -0.783124811, x3 = -0.782808234, x4 = x5 = -0.782808123; if x1 = 0.6, then x2 = 0.568003853, x3 = x4 = 0.568025739 ; if x1 = 2, then x2 = 1.979461151, x3 = 1.979019264, x4 = x5 = 1.979019061 17. The graphs of y = x2 and y = 2x + 1 intersect at points near x = -0.5 and x = 1; x2 = 2x + 1, x4 - 2x - 1 = 0. Let f (x) = x4 - 2x - 1, then f (x) = 4x3 - 2 so xn+1 = xn - x4 - 2xn - 1 n . 4x3 - 2 n
0.5 0 4
1
If x1 = -0.5, then x2 = -0.475, x3 = -0.474626695, x4 = x5 = -0.474626618; if x1 = 1, then x2 = 2, x3 = 1.633333333, . . . , x8 = x9 = 1.395336994. 18. The graphs of y = 1 x3 - 1 and y = cos x - 2 intersect 8 when x = 0 and near the point x = -2. Let f (x) = 1 3 3 2 8 x + 1 - cos x so that f (x) = 8 x + sin x. Then xn+1 = xn - x3 /8 + 1 - cos x . 3x2 /8 + sin x
2
0 3 2
If x1 = -2 then x2 = -2.70449471, x3 = -2.46018026, . . . , x6 = x7 = -2.40629382
3
Exercise Set 5.6
207
19. The graphs of y = 1 and y = ex sin x intersect near the points x = 1 and x = 3. Let f (x) = 1-ex sin x, f (x) = -ex (cos x + sin x), and xn+1 = xn + 1 - ex sin x . ex (cos x + sin x) If x1 = 1 then
0
8
x2 = 0.65725814, x3 = 0.59118311, . . . , x5 = x6 = 0.58853274, and if x1 = 3 then x2 = 3.10759324, x3 = 3.09649396, . . . , x5 = x6 = 3.09636393. 20. The graphs of y = e-x and y = ln x intersect near x = 1.3; let f (x) = e-x - ln x, f (x) = -e-x - 1/x, x1 = 1.3, xn+1 = xn + - ln xn e , x2 = 1.309759929, -xn + 1/x e n
-xn
3.2 0
1 0 2
x4 = x5 = 1.309799586
4
21. (a) f (x) = x2 - a, f (x) = 2x, xn+1 = xn -
x2 - a 1 n = 2xn 2
xn +
a xn
(b) a = 10; x1 = 3, x2 = 3.166666667, x3 = 3.162280702, x4 = x5 = 3.162277660 22. (a) f (x) = 1 1 - a, f (x) = - 2 , xn+1 = xn (2 - axn ) x x
(b) a = 17; x1 = 0.05, x2 = 0.0575, x3 = 0.058793750, x5 = x6 = 0.058823529 23. f (x) = x3 + 2x - 5; solve f (x) = 0 to find the critical points. Graph y = x3 and y = -2x + 5 to see that they intersect at a point near x = 1.25; f (x) = 3x2 + 2 so xn+1 = xn - x3 + 2xn - 5 n . 3x2 + 2 n
x1 = 1.25, x2 = 1.3317757009, x3 = 1.3282755613, x4 = 1.3282688557, x5 = 1.3282688557 so the minimum value of f (x) occurs at x 1.3282688557 because f (x) > 0; its value is approximately -4.098859123. 24. From a rough sketch of y = x sin x we see that the maximum occurs at a point near x = 2, which will be a point where f (x) = x cos x + sin x = 0. f (x) = 2 cos x - x sin x so xn+1 = xn - xn cos xn + sin xn xn + tan xn = xn - . 2 cos xn - xn sin xn 2 - xn tan xn
x1 = 2, x2 = 2.029048281, x3 = 2.028757866, x4 = x5 = 2.028757838; the maximum value is approximately 1.819705741. 25. A graphing utility shows that there are two inflection points at x 0.25, -1.25. These points e-x are the zeros of f (x) = (x4 + 4x3 + 8x2 + 4x - 1) 2 . It is equivalent to find the zeros of (x + 1)3 4 3 2 g(x) = x +4x +8x +4x-1. One root is x = -1 by inspection. Since g (x) = 4x3 +12x2 +16x+4, Newton's Method becomes xn+1 = xn - x4 + 4x3 + 8x2 + 4xn - 1 n n n 4x3 + 12x2 + 16xn + 4 n n
With x0 = 0.25, x1 = 0.18572695, x2 = 0.179563312, x3 = 0.179509029, x4 = x5 = 0.179509025. So the points of inflection are at x 0.18, x = -1.
208
Chapter 5
26. f (x) = -2 tan-1 x +
1 - 2x = 0 for x = x1 0.2451467013, f (x1 ) 0.1225363521 x2 + 1
27. Let f (x) be the square of the distance between (1, 0) and any point (x, x2 ) on the parabola, then f (x) = (x - 1)2 + (x2 - 0)2 = x4 + x2 - 2x + 1 and f (x) = 4x3 + 2x - 2. Solve f (x) = 0 to find 4x3 + 2xn - 2 2x3 + xn - 1 n = xn - n 2 . 2 +2 12xn 6xn + 1 x1 = 1, x2 = 0.714285714, x3 = 0.605168701, . . . , x6 = x7 = 0.589754512; the coordinates are approximately (0.589754512, 0.347810385). the critical points; f (x) = 12x2 + 2 so xn+1 = xn - 28. The area is A = xy = x cos x so dA/dx = cos x - x sin x. Find x so that dA/dx = 0; cos xn - xn sin xn 1 - xn tan xn = xn + . 2 sin xn + xn cos xn 2 tan xn + xn x1 = 1, x2 = 0.864536397, x3 = 0.860339078, x4 = x5 = 0.860333589; y 0.652184624. d2 A/dx2 = -2 sin x - x cos x so xn+1 = xn + 29. (a) Let s be the arc length, and L the length of the chord, then s = 1.5L. But s = r and L = 2r sin(/2) so r = 3r sin(/2), - 3 sin(/2) = 0. n - 3 sin(n /2) (b) Let f () = - 3 sin(/2), then f () = 1 - 1.5 cos(/2) so n+1 = n - . 1 - 1.5 cos(n /2) 1 = 3, 2 = 2.991592920, 3 = 2.991563137, 4 = 5 = 2.991563136 rad so 171 . 30. r2 ( - sin )/2 = r2 /4 so - sin - /2 = 0. Let f () = - sin - /2, then f () = 1 - cos so n+1 = n - sin n - /2 . 1 - cos n
1 = 2, 2 = 2.339014106, 3 = 2.310063197, . . . , 5 = 6 = 2.309881460 rad; 132 . 31. If x = 1, then y 4 + y = 1, y 4 + y - 1 = 0. Graph z = y 4 and z = 1 - y to see that they intersect near y = -1 and y = 1. Let f (y) = y 4 + y - 1, then f (y) = 4y 3 + 1 so yn+1 = yn -
4 yn + yn - 1 . 3 4yn + 1
If y1 = -1, then y2 = -1.333333333, y3 = -1.235807860, . . . , y6 = y7 = -1.220744085; if y1 = 1, then y2 = 0.8, y3 = 0.731233596, . . . , y6 = y7 = 0.724491959. 32. If x = 1, then 2y - cos y = 0. Graph z = 2y and z = cos y to see that they intersect near y = 0.5. 2yn - cos yn Let f (y) = 2y - cos y, then f (y) = 2 + sin y so yn+1 = yn - . 2 + sin yn y1 = 0.5, y2 = 0.450626693, y3 = 0.450183648, y4 = y5 = 0.450183611. 33. S(25) = 250,000 = 5000 (1 + i)25 - 1 ; set f (i) = 50i - (1 + i)25 + 1, f (i) = 50 - 25(1 + i)24 ; solve i f (i) = 0. Set i0 = .06 and ik+1 = ik - 50i - (1 + i)25 + 1 / 50 - 25(1 + i)24 . Then i1 = 0.05430, i2 = 0.05338, i3 = 0.05336, . . . , i = 0.053362.
0.5
34. (a) x1 = 2, x2 = 5.3333, x3 = 11.055, x4 = 22.293, x5 = 44.676
0 0
15
(b) x1 = 0.5, x2 = -0.3333, x3 = 0.0833, x4 = -0.0012, x5 = 0.0000 (and xn = 0 for n 6)
Exercise Set 5.7
209 x2 -0.7500 x3 0.2917 x4 -1.5685 x5 -0.4654 x6 0.8415 x7 -0.1734 x8 2.7970 x9 1.2197 x10 0.1999
35. (a)
x1 0.5000
(b) The sequence xn must diverge, since if it did converge then f (x) = x2 + 1 = 0 would have a solution. It seems the xn are oscillating back and forth in a quasi-cyclical fashion. 36. (a) xn+1 = xn , i.e. the constant sequence xn is generated. (b) This is equivalent to f (xn ) = 0 as in part (a). (c) xn = xn+2 = xn+1 - f (xn+1 ) f (xn ) f (xn+1 ) f (xn+1 ) = xn - - , so f( xn+1 ) = - f (xn ) f (xn+1 ) f (xn ) f (xn+1 ) f (xn )
37. (a) |xn+1 - xn | |xn+1 - c| + |c - xn | < 1/n + 1/n = 2/n (b) The closed interval [c - 1, c + 1] contains all of the xn , since |xn - c| < 1/n. Let M be an upper bound for |f (x)| on [c - 1, c + 1]. Since xn+1 = xn - f (xn )/f (xn ) it follows that |f (xn )| |f (xn )||xn+1 - xn | < M |xn+1 - xn | < 2M/n. (c) Assume that f (c) = 0. The sequence xn converges to c, since |xn - c| < 1/n. By the continuity of f , f (c) = f ( lim xn ) = lim f (xn ). Let = |f (c)|/2. Choose N such that |f (xn ) - f (c)| < /2 for n > N . Then |f (xn ) - f (c)| < |f (c)|/2 for n > N , so -|f (c)|/2 < f (xn ) - f (c) < |f (c)/2|. If f (c) > 0 then f (xn ) > f (c) - |f (c)|/2 = f (c)/2. If f (c) < 0, then f (xn ) < f (c) + |f (c)|/2 = -|f (c)|/2, or |f (xn )| > |f (c)|/2. (d) From (b) it follows that
n+ n+ n+ n+
lim f (xn ) = 0.
From (c) it follows that if f (c) = 0 then
lim f (xn ) = 0, a contradiction. The conclusion, then, is that f (c) = 0.
EXERCISE SET 5.7
1. f (3) = f (5) = 0; f (x) = 2x - 8, 2c - 8 = 0, c = 4, f (4) = 0 2. f (0) = f (2) = 0, f (x) = 3x - 6x + 2, 3c - 6c + 2 = 0; c =
2 2
6
36 - 24 = 1 3/3 6
3. f (/2) = f (3/2) = 0, f (x) = - sin x, - sin c = 0, c = 4. f (-1) = f (3) = 0; f (1) = 0; f (x) = 2(1 - x)/(4 + 2x - x2 ); 2(1 - c) = 0, c = 1 5. f (0) = f (4) = 0, f (x) = 1 1 1 1 - , - = 0, c = 1 2 2 x 2 2 c 2 4 2 4 + 2 , - 3 + 2 = 0, -6 + 4c = 0, c = 3/2 3 x 3x c 3c
6. f (1) = f (3) = 0, f (x) = -
7. (f (5) - f (-3))/(5 - (-3)) = 1; f (x) = 2x - 1; 2c - 1 = 1, c = 1 8. f (-1) = -6, f (2) = 6, f (x) = 3x2 + 1, 3c2 + 1 = c = 1 is in (-1, 2) 6 - (-6) = 4, c2 = 1, c = 1 of which only 2 - (-1)
1 1 1 2-1 = , c + 1 = 3/2, c + 1 = 9/4, c = 5/4 9. f (0) = 1, f (3) = 2, f (x) = , = 3-0 3 2 x+1 2 c+1 = 10. f (4) = 15/4, f (3) = 8/3, solve f (c) = (15/4-8/3)/1 = 13/12; f (x) 1+1/x2 , f (c) = 1+1/c2 = 13/12, c2 = 12, c = 2 3, but -2 3 is not in the interval, so c = 2 3.
210
Chapter 5
1 x c 4-0 = , -2c = 25 - c2 , 11. f (-5) = 0, f (3) = 4, f (x) = - , - = 2 2 3 - (-5) 2 25 - x 25 - c 4c2 = 25 - c2 , c2 = 5, c = - 5 (we reject c = 5 because it does not satisfy the equation -2c = 25 - c2 ) 12. f (2) = 1, f (5) = 1/4, f (x) = -1/(x - 1)2 , - c = -1 (reject), or c = 3 13. (a) f (-2) = f (1) = 0 The interval is [-2, 1] 1 1 1/4 - 1 = - , (c - 1)2 = 4, c - 1 = 2, = (c - 1)2 5-2 4 (b) c = -1.29
6
2 2
1
(c) x0 = -1, x1 = -1.5, x2 = -1.32, x3 = -1.290, x4 = -1.2885843 14. (a) m = 0+3 f (-2) - f (1) = = -1 so y + 3 = -(x - 1), y = -x - 2 -2 - 1 -3
(b) f (x) = 3x2 - 4 = -1 has solutions x = 1; discard x = 1, so c = -1 (c) y - (3) = -(x - (-1)) or y = -x + 2 (d)
4
3
2
4
15. (a) f (x) = sec2 x, sec2 c = 0 has no solution 16. (a) f (-1) = 1, f (8) = 4, f (x) = 2 -1/3 x 3
(b) tan x is not continuous on [0, ]
1 2 -1/3 4-1 c = , c1/3 = 2, c = 8 which is not in (-1, 8). = 3 8 - (-1) 3 (b) x2/3 is not differentiable at x = 0, which is in (-1, 8). 17. (a) Two x-intercepts of f determine two solutions a and b of f (x) = 0; by Rolle's Theorem there exists a point c between a and b such that f (c) = 0, i.e. c is an x-intercept for f . (b) f (x) = sin x = 0 at x = n, and f (x) = cos x = 0 at x = n + /2, which lies between n and (n + 1), (n = 0, 1, 2, . . .) 18. f (x1 ) - f (x0 ) is the average rate of change of y with respect to x on the interval [x0 , x1 ]. By x1 - x0 the Mean-Value Theorem there is a value c in (x0 , x1 ) such that the instantaneous rate of change f (c) = f (x1 ) - f (x0 ) . x1 - x0
Exercise Set 5.7
211
19. Let s(t) be the position function of the automobile for 0 t 5, then by the Mean-Value Theorem there is at least one point c in (0, 5) where s (c) = v(c) = [s(5) - s(0)]/(5 - 0) = 4/5 = 0.8 mi/min = 48 mi/h. 20. Let T (t) denote the temperature at time with t = 0 denoting 11 AM, then T (0) = 76 and T (12) = 52. (a) By the Mean-Value Theorem there is a value c between 0 and 12 such that T (c) = [T (12) - T (0)]/(12 - 0) = (52 - 76)/(12) = -2 F/h. (b) Assume that T (t1 ) = 88 F where 0 < t1 < 12, then there is at least one point c in (t1 , 12) where T (c) = [T (12)-T (t1 )]/(12-t1 ) = (52-88)/(12-t1 ) = -36/(12-t1 ). But 12-t1 < 12 so T (c) < -3 F. 21. Let f (t) and g(t) denote the distances from the first and second runners to the starting point, and let h(t) = f (t) - g(t). Since they start (at t = 0) and finish (at t = t1 ) at the same time, h(0) = h(t1 ) = 0, so by Rolle's Theorem there is a time t2 for which h (t2 ) = 0, i.e. f (t2 ) = g (t2 ); so they have the same velocity at time t2 . 22. f (x) = x - (2 - x) ln(2 - x); solve f (x) = 0 in (0, 1). f (0) = -2 ln 2 < 0; f (1) = 1 > 0. By the Intermediate-Value Theorem (Theorem 2.5.7) there exists x in (0, 1) such that f (x) = 0. 23. (a) By the Constant Difference Theorem f (x) - g(x) = k for some k; since f (x0 ) = g(x0 ), k = 0, so f (x) = g(x) for all x. (b) Set f (x) = sin2 x + cos2 x, g(x) = 1; then f (x) = 2 sin x cos x - 2 cos x sin x = 0 = g (x). Since f (0) = 1 = g(0), f (x) = g(x) for all x. 24. (a) By the Constant Difference Theorem f (x) - g(x) = k for some k; since f (x0 ) - g(x0 ) = c, k = c, so f (x) - g(x) = c for all x. (b) Set f (x) = (x - 1)3 , g(x) = (x2 + 3)(x - 3). Then f (x) = 3(x - 1)2 , g (x) = (x2 + 3) + 2x(x - 3) = 3x2 - 6x + 3 = 3(x2 - 2x + 1) = 3(x - 1)2 , so f (x) = g (x) and hence f (x) - g(x) = k. Expand f (x) and g(x) to get h(x) = f (x) - g(x) = (x3 - 3x2 + 3x - 1) - (x3 - 3x2 + 3x - 9) = 8. (c) h(x) = x3 - 3x2 + 3x - 1 - (x3 - 3x2 + 3x - 9) = 8 25. By the Constant Difference Theorem it follows that f (x) = g(x) + c; since g(1) = 0 and f (1) = 2 we get c = 2; f (x) = xex - ex + 2. 26. By the Constant Difference Theorem f (x) = tan-1 x + C and 2 = f (1) = tan-1 (1) + C = /4 + C, C = 2 - /4, f (x) = tan-1 x + 2 - /4. 27. (a) If x, y belong to I and x < y then for some c in I, f (y) - f (x) = f (c), y-x so |f (x) - f (y)| = |f (c)||x - y| M |x - y|; if x > y exchange x and y; if x = y the inequality also holds.
(b) f (x) = sin x, f (x) = cos x, |f (x)| 1 = M , so |f (x) - f (y)| |x - y| or | sin x - sin y| |x - y|. 28. (a) If x, y belong to I and x < y then for some c in I, f (y) - f (x) = f (c), y-x so |f (x) - f (y)| = |f (c)||x - y| M |x - y|; if x > y exchange x and y; if x = y the inequality also holds.
(b) If x and y belong to (-/2, /2) and f (x) = tan x, then |f (x)| = sec2 x 1 and | tan x - tan y| |x - y| (c) y lies in (-/2, /2) if and only if -y does; use Part (b) and replace y with -y
212
Chapter 5
29. (a) Let f (x) = x. By the Mean-Value Theorem there is a number c between x and y such that y- x 1 1 y-x = < for c in (x, y), thus y - x < y-x 2 c 2 x 2 x 1 (b) multiply through and rearrange to get xy < (x + y). 2 30. Suppose that f (x) has at least two distinct real solutions r1 and r2 in I. Then f (r1 ) = f (r2 ) = 0 so by Rolle's Theorem there is at least one number between r1 and r2 where f (x) = 0, but this contradicts the assumption that f (x) = 0, so f (x) = 0 must have fewer than two distinct solutions in I. 31. (a) If f (x) = x3 + 4x - 1 then f (x) = 3x2 + 4 is never zero, so by Exercise 30 f has at most one real root; since f is a cubic polynomial it has at least one real root, so it has exactly one real root. (b) Let f (x) = ax3 + bx2 + cx + d. If f (x) = 0 has at least two distinct real solutions r1 and r2 , then f (r1 ) = f (r2 ) = 0 and by Rolle's Theorem there is at least one number between r1 and r2 where f (x) = 0. But f (x) = 3ax2 + + c = 0 for 2bx x = (-2b 4b2 - 12ac)/(6a) = (-b b2 - 3ac)/(3a), which are not real if b2 - 3ac < 0 so f (x) = 0 must have fewer than two distinct real solutions. 1 1 1 4- 3 1 1 32. f (x) = , = = 2 - 3. But < < for c in (3, 4) so 4-3 4 2 x 2 c 2 c 2 3 1 1 < 2 - 3 < , 0.25 < 2 - 3 < 0.29, -1.75 < - 3 < -1.71, 1.71 < 3 < 1.75. 4 2 3 33. By the Mean-Value Theorem on the interval [0, x], tan-1 x 1 tan-1 x - tan-1 0 = = for c in (0, x), but x-0 x 1 + c2 x 1 1 tan-1 x 1 < 1, < < 1 for c in (0, x) so < < tan-1 x < x. 1 + x2 1 + c2 1 + x2 x 1 + x2 34. (a) d 2 [f (x) - g 2 (x)] = 2f (x)f (x) - 2g(x)g (x) = 2f (x)g(x) - 2g(x)f (x) = 0, so f 2 - g 2 is dx constant.
(b) f (x) = 1 (ex - e-x ) = g(x), g (x) = 1 (ex + e-x ) = f (x) 2 2 35. (a) d 2 [f (x) + g 2 (x)] = 2f (x)f (x) + 2g(x)g (x) = 2f (x)g(x) + 2g(x)[-f (x)] = 0, dx so f 2 (x) + g 2 (x) is constant.
(b) f (x) = sin x and g(x) = cos x 36. Let h = f - g, then h is continuous on [a, b], differentiable on (a, b), and h(a) = f (a) - g(a) = 0, h(b) = f (b) - g(b) = 0. By Rolle's Theorem there is some c in (a, b) where h (c) = 0. But h (c) = f (c) - g (c) so f (c) - g (c) = 0, f (c) = g (c). 37.
y
]
x c
Exercise Set 5.8
213
38. (a) Suppose f (x) = 0 more than once in (a, b), say at c1 and c2 . Then f (c1 ) = f (c2 ) = 0 and by using Rolle's Theorem on f , there is some c between c1 and c2 where f (c) = 0, which contradicts the fact that f (x) > 0 so f (x) = 0 at most once in (a, b). (b) If f (x) > 0 for all x in (a, b), then f is concave up on (a, b) and has at most one relative extremum, which would be a relative minimum, on (a, b). 39. (a) similar to the proof of Part (a) with f (c) < 0 (b) similar to the proof of Part (a) with f (c) = 0 40. Let x = x0 be sufficiently near x0 so that there exists (by the Mean-Value Theorem) a number c (which depends on x) between x and x0 , such that f (x) - f (x0 = f (c). x - x0 Since c is between x and x0 it follows that f (x0 ) = lim
xx0 xx0 xx0
f (x) - f (x0 ) x - x0
(by definition of derivative) (by the Mean-Value Theorem) (since lim f (x) exists and c is between x and x0 ).
= lim f (c) = lim f (x)
41. If f is differentiable at x = 1, then f is continuous there;
x1+
lim f (x) = lim- f (x) = f (1) = 3, a + b = 3; lim+ f (x) = a and
x1 x1
x1-
lim f (x) = 6 so a = 6 and b = 3 - 6 = -3. lim f (x) = lim 2x = 0 and lim f (x) = lim 2x = 0; f (0) does not exist because f is
x0- x0+ x0+
42. (a) (b)
x0-
not continuous at x = 0.
x0
x0- x0-
lim f (x) = lim+ f (x) = 0 and f is continuous at x = 0, so f (0) = 0; lim f (x) = lim (2) = 2 and lim f (x) = lim 6x = 0, so f (0) does not exist.
x0- x0+ x0+
43. From Section 3.2 a function has a vertical tangent line at a point of its graph if the slopes of secant lines through the point approach + or -. Suppose f is continuous at x = x0 and lim+ f (x) = +. Then a secant line through (x1 , f (x1 )) and (x0 , f (x0 )), assuming x1 > x0 , will
xx0
have slope
f (x1 ) - f (x0 ) . By the Mean Value Theorem, this quotient is equal to f (c) for some x1 - x0
cx0
c between x0 and x1 . But as x1 approaches x0 , c must also approach x0 , and it is given that lim f (c) = +, so the slope of the secant line approaches +. The argument can be altered +
appropriately for x1 < x0 , and/or for f (c) approaching -.
EXERCISE SET 5.8
1. (a) positive, negative, slowing down (c) negative, positive, slowing down 2. (a) positive, slowing down (c) positive, speeding up (b) positive, positive, speeding up
(b) negative, slowing down
214
Chapter 5
3. (a) (b) (c) (d)
left because v = ds/dt < 0 at t0 negative because a = d2 s/dt2 and the curve is concave down at t0 (d2 s/dt2 < 0) speeding up because v and a have the same sign v < 0 and a > 0 at t1 so the particle is slowing down because v and a have opposite signs. 5. s (m)
4. (a) III (b) I (c) II
t (s)
6. (a) when s 0, so 0 < t < 2 and 4 < t 8 (c) when s is decreasing, so 0 t < 3 7.
15 10 5 0 5 10 t 1 2 3 4 5 6 |v|
(b) when the slope is zero, at t = 3
15
a
t 6
15
8. (a) v (30 - 10)/(15 - 10) = 20/5 = 4 m/s (b)
v a
t 25 (1) (2) 25
t
9. (a) At 60 mi/h the tangent line seems...