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Chapter 4 25

# Chapter 4 25 - SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE...

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Unformatted text preview: SECTION 4.3 HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH 287 32. (a) f is increasing where f’ is positive. on (1. 6) and (8. 00). and decreasing where f’ is negative. on (O7 1) and (6. 8). (b) f has a local maximum where f’ changes from positive to negative. at ac = 6. and local minima where f’ changes from negative to positive. at m = 1 and at m = 8. (c) f is concave upward where f’ is increasing. that is. on (0. 2). (3. 5). (e) and (7.00). and concave downward where f’ is decreasing. that is. on (2.3) and (5.7). (d) There are points of inﬂection where f changes its direction of concavity. at ac : 2. x : 3, z : 5 and a; : 7. 33. (a) f(a:) : 23:3 — 3m? — 12:1: ;» f’(a:) = 6x2 — 6:1: — 12 = 6(ac2 — 3: ~ 2) : 6(a: ~ 2)(:c + 1). f'(a:) > 0 41> ac < —10r\$ > 2 and f'(a:) < 0 4:} —1 < a: < 2. So f is increasing on (700, *1) and (2.00). and f is decreasing on (—1.2). (b) Since f changes from increasing to decreasing at av : —1, f(71) : 7 is a local maximum value. Since f changes from decreasing to increasing at :1: : 2. f(2) : —20 is a local minimum value. (c) f"(a:) : 6(2m — 1) :> f"(ar) > 0 on (g. 00) and Wm < 0 on (d) (—00, g). So f is concave upward on (g. 00) and concave (—1. 7) y downward on (foo. %) There is a change in concavity at m : % and we have an inﬂection point at (é. —%). 34. (a) ﬂat) : 2 + 3x ~ 903 => f’(ac) 2 3 ~ 3w2 2 —3(m2 — 1): —3(\$ +1)(m — 1). f'(m) > 0 (I) —1 < r < 1 and f’(:v) < 0 <:> :1: < —1 orm > 1. So f is increasing on (~1. 1) and f is decreasing on (~oo.—1) and (1.00). (b) f(—1) : 0 is a local minimum value and f(1) = 4 is a local (d) maximum value. (c) f"(av) = —6a: => f"(:c) > 0 on (~00, O) and f"(ar) < 0 on (0., 00) . So f is concave upward on (—00. 0) and concave downward on (0. 00). There is an inﬂection point at (0, 2). 35. (a) f(\$) : 2:4 ~ 69:2 => f’(:z:) : 4m3 — 12m : 433(562 — 3): 0 when :r = 0. ::\/§. Interval 4a: m2 — 3 f'(m) f x < ,\/3_, — + — decreasing on (—00. ﬁx/g) *x/g < a: < 0 — — —— increasing on (~x/g, 0) 0 < :1: < x/I; + — ~ decreasing on (0. M3) an > x/i’: + + —— increasing on (M3: 00) ...
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