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

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

Graphs of Functions and Derivatives 159

(e) A sketch of the graph offis shown in Figure 8.4-3.

rel. max.

rel. min.

y

x

rel. max.

pts. of

inflection

• 4
  
  
  • 3
  
  
  
  


• 20 2 4

Figure 8.4-3

Example 3

Given the graph of f′in Figure 8.4-4, find where the functionf (a) has a horizontal tan-

gent, (b) has its relative extrema, (c) is increasing or decreasing, (d) has a point of inflection,

and (e) is concave upward or downward.

• 2
  
  
  • 2
  
  
  
  


• 5 – 4 – 3


• 3


• 1


• 1

01 3 5 46789 x

y

1

2

3

4

f′

2

Figure 8.4-4
(a) f′(x)=0atx=−4, 2, 4, 8. Thus,fhas a horizontal tangent at these values.
(b) Summarize the information off′on a number line:

• 42 8

0000

4

f incr.

f′

decr. incr. decr. incr.

++––+

The First Derivative Test indicates that fhas relative maximums atx=−4 and 4; and

fhas relative minimums atx=2 and 8.