124 STEP 4. Review the Knowledge You Need to Score High
Example 2
Given the graph of f′in Figure 7.4-2, 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 –2 0 2 4
f′
Figure 7.4-2
(a) Summarize the information of f′on a number line:
–4 0
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 of f′′on a number line:
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,∞).