5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 17:21

164 STEP 4. Review the Knowledge You Need to Score High

6. Find the values ofxwhere f′is concave upward. (See Figure 8.5-8.)
   y

x

f′′

• 30 3

4

Figure 8.5-8

Answer: f′is concave upward on (−∞, 0). (See Figure 8.5-9.)

f′′ incr. decr.

+ –

concave

upward

concave

downward

x

f′′′

f′

0

Figure 8.5-9

8.6 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 the interval [0, 1] and find all values ofc
   that 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. Given f(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 8.6-1,
   identify the points where:
   (a) f′<0 andf′′>0.
   (b) f′<0 andf′′<0.
   (c) f′=0.
   (d) f′′does not exist.

y

f

x

A

B

0

C

D

E

Figure 8.6-1