116 STEP 4. Review the Knowledge You Need to Score High
Note that if a point (a,f(a)) is a point of inflection, thenf′′(c)=0orf′′(c) does not
exist. (The converse of the statement is not necessarily true.)
Note: There are some textbooks that define a point of inflection as a point where the
concavity changes and do not require the existence of a tangent at the point of inflection.
In that case, the point at the cusp in Figure 7.2-18 would be a point of inflection.
Example 1
The graph off′, the derivative of a functionf, is shown in Figure 7.2-19. Find the points
of inflection off and determine where the function f is concave upward and where it is
concave downward on [−3, 5].
1
1
–1
–2
–3
2
3
4
–1–2–3^02345
y
x
f′
Figure 7.2-19
Solution: (See Figure 7.2-20.)
–3 0 3 5
incr. decr. incr.
Concave
Upward
Concave
Upward
Concave
Downard
pt. of
infl.
pt. of
infl.
+–+
x
f ′
f′′
f
Figure 7.2-20
Thus,fis concave upward on [−3, 0) and (3, 5], and is concave downward on (0, 3).
There are two points of inflection: one atx=0 and the other atx=3.