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

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

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

Example 2

Given the graph of f′in Figure 8.4-2, find where the functionf: (a) has its relative

maximum(s) or relative minimums, (b) is increasing or decreasing, (c) has its point(s) of

inflection, (d) is concave upward or downward, and (e) iff(−2)= f(2)=1 and f(0)=−3,

draw a sketch off.

x

y

– 4 – (^2024)
f′
Figure 8.4-2
(a) Summarize the information of f′on a number line:

• 40

00 0

4

f incr. incr.

f′

decr. decr.

+ – + –

The function fhas a relative maximum atx=−4 and atx=4, and a relative minimum

atx=0.

(b) The function f is increasing on interval (−∞,−4] and [0, 4], and f is decreasing on

[−4, 0] and [4,∞).

(c) Summarize the information off′′on a number line:

concave

downward

concave

downward

0

• 22
  
  
  • concave
    upward
  
  
  
  

decr. incr. decr.

––

f′

f′′

f

A change of concavity occurs atx=−2 and atx=2 and f′exists atx=−2 and at

x=2, which implies that there is a tangent line to the graph offatx=−2 and atx=2.

Therefore, f has a point of inflection atx=−2 and atx=2.

(d) The graph of f is concave upward on the interval (−2, 2) and concave downward on

(−∞,−2) and (2,∞).