5 Steps to a 5 AP Calculus BC 2019

118 STEP 4. Review the Knowledge You Need to Score High

Step 4: Set up a table.

INTERVALS (−∞,2) x= 2 (2,∞)

Test Point 0 5

f′′(x) − 0 +

f(x) concave point of concave

downward inflection upward

Since f(x) has change of concavity atx=2, the point (2, f(2)) is a point of

inflection. f(2)=(2)^3 −6(2)^2 +12(2)− 8 =0.

Step 5: Write a conclusion.

Thus,f(x) is concave downward on (−∞, 2), concave upward on (2,∞) andf(x)

has a point of inflection at (2, 0). (See Figure 7.2-23.)

[–1,5] by [–5,5]

Figure 7.2-23

Example 4

Find the points of inflection of f(x)=(x−1)^2 /^3 and determine the intervals where the

functionfis concave upward and where it is concave downward.

Step 1: Find f′(x) andf′′(x).

f′(x)=

2

3

(x−1)−^1 /^3 =

2

3(x−1)^1 /^3

f′′(x)=−

2

9

(x−1)−^4 /^3 =

− 2

9(x−1)^4 /^3

Step 2: Find all values ofxwhere f′′(x)=0orf′′(x) is undefined.

Note that f′′(x)/=0 and that f′′(1) is undefined.

Step 3: Determine intervals.

The intervals are (−∞, 1), and (1,∞).