NAME: Brief Answers
TEST4T/MAC2311
Page 1 of 5
Read Me First:
Show all essential work very neatly. Use
correct notation when presenting your computations and arguments.
Write using complete sentences. Be careful.
Remember this: "="
denotes "equals" , "
⇒
" denotes "implies" , and "
⇔
" denotes "is
equivalent to".
Do not "box" your answers. Communicate. Show
me all the magic on the page.
1. (16 pts.)
Fill in the blanks of the following analysis with
the correct terminology.
Let f(x) = x
4
 8x
3
. Then f
′
(x) = 4x
3
 24x
2
= 4(x  0)
2
(x  6).
Consequently, x = 0 and x = 6 are critical (stationary)
points of
f. Since f
′
(x) > 0 for 6 < x, f is increasing
on the
set (6,
∞
). Also, because f
′
(x) < 0 when 0 < x < 6 or x < 0,
and f is continuous, f is decreasing
on the interval
(
∞
, 6). Using the first derivative test, it follows that f
has a(n) relative minimum
at x = 6, and
neither a relative max nor a relative min
at x = 0.
Since f
″
(x) = 12x
2
 48x = 12x (x  4), we have f
″
(0) = 0,
f
″
(4) = 0, f
″
(x) < 0 when 0 < x < 4, and f
″
(x) > 0 when x > 4 or
x < 0. Thus, f is concave down
on the interval
(0,4), f is concave up
on the set (
∞
,0)
∪
(4,
∞
),
and f has inflection points
at x = 0 and x = 4.
2. (4 pts.)
Rolle’s Theorem states that if f(x) is continuous
on [a,b] with f(a) = f(b) = 0 and differentiable on (a,b), then
there is a number c in (a,b) such that f
′
(c) = 0.
Give an
example of a function f(x) defined on [1,1] with f
differentiable on (1,1) and f(1) = f(1) = 0 but such that there
is no number c in (1,1) with f
′
(c) = 0.
[Hint: Which hypothesis
above must you violate??]
//
A suitable example clearly must fail to be continuous on the
interval [1,1] and yet satisfy the remaining hypotheses. Here
is one such, defined in pieces, of course.
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 Fall '08
 STAFF
 Calculus, $1, $2.00, $2, $1.00, 2 feet

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