5 Steps to a 5 AP Calculus BC 2019

Graphs of Functions and Derivatives 119

Step 4: Set up a table.

INTERVALS (−∞,1) x= 1 (1,∞)

Test Point 0 2

f′′(x) − undefined −

f(x) concave no change concave

downward of cancavity downward

Note that sincef(x) has no change of concavity atx=1,f does not have a point

of inflection.

Step 5: Write a conclusion.
Therefore, f(x) is concave downward on (−∞,∞) and has no point of inflection.
(See Figure 7.2-24.)

[–3,5] by [–1,4]

Figure 7.2-24

Example 5

The graph of f is shown in Figure 7.2-25 and f is twice differentiable. Which of the
following statements is true?

y

f

x

05

Figure 7.2-25

(A) f(5)< f′(5)< f′′(5)

(B) f′′(5)< f′(5)< f(5)

(C) f′(5)<f(5)< f′′(5)

(D) f′(5)<f′′(5)< f(5)