5 Steps to a 5 AP Calculus BC 2019

Graphs of Functions and Derivatives 137

7.7 Practice Problems

Part A The use of a calculator is not

allowed.

1. Iff(x)=x^3 −x^2 − 2 x, show that the
   hypotheses of Rolle’s Theorem are satisfied
   on the interval [−1, 2] and find all values of
   cthat satisfy the conclusion of the theorem.


2. Letf(x)=ex. Show that the hypotheses of
   the Mean Value Theorem are satisfied on
   [0, 1] and find all values ofcthat satisfy the
   conclusion of the theorem.


3. Determine the intervals in which the graph
   off(x)=
   x^2 + 9
   x^2 − 25
   is concave upward or
   downward.


4. Givenf(x)=x+sinx 0 ≤x≤ 2 π, find
   all points of inflection off.


5. Show that the absolute minimum of
   f(x)=

√

25 −x^2 on [−5, 5] is 0 and the

absolute maximum is 5.

6. Given the functionfin Figure 7.7-1,
   identify the points where:
   (a) f′<0 and f′′>0, (b) f′<0 and f′′<0,
   (c) f′=0, (d) f′′does not exist.


7. Given the graph off′′in Figure 7.7-2,
   determine the values ofxat which the
   functionfhas a point of inflection. (See
   Figure 7.7-2.)


8. Iff′′(x)=x^2 (x+3)(x−5), find the values
   ofxat which the graph of fhas a change of
   concavity.


9. The graph off′on [−3, 3] is shown in
   Figure 7.7-3. Find the values ofxon
   [−3, 3] such that (a)fis increasing and
   (b)f is concave downward.

y

f

x

A

B

0

C

D

E

Figure 7.7-1

Figure 7.7-2

y

–3 3210 x

f ′

Figure 7.7-3