5 Steps to a 5 AP Calculus BC 2019

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.