Graphs of Functions and Derivatives 125
(e) A sketch of the graph of fis shown in Figure 7.4-3.
Figure 7.4-3
Example 3
Given the graph of f′in Figure 7.4-4, 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 3467895 x
y
1
2
3
4
f ′
2
Figure 7.4-4
(a) f′(x)=0atx=−4, 2, 4, 8. Thus,fhas a horizontal tangent at these values.
(b) Summarize the information of f′on a number line:
The First Derivative Test indicates that fhas relative maximums atx=−4 and 4; and
fhas relative minimums atx=2 and 8.