5 Steps to a 5 AP Calculus BC 2019

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

Example 4

Find the open intervals on whichf(x)=(x^2 −9)^2 /^3 is increasing or decreasing.

Step 1: Find the critical numbers of f.

f′(x)=

2

3

(x^2 −9)−^1 /^3 (2x)=

4 x

3(x^2 −9)^1 /^3

Setf′(x)= 0 ⇒ 4 x=0orx=0.

Since f′(x) is a rational function,f′(x) is undefined at values where the denomi-

nator is 0. Thus, setx^2 − 9 = 0 ⇒x=3orx=−3. Therefore, the critical numbers

are−3, 0, and 3.

Step 2: Determine intervals.

Intervals are (−∞,−3), (−3, 0), (0, 3), and (3,∞).

Step 3: Set up a table.

INTERVALS (−∞,−3) (−3, 0) (0, 3) (3,∞)

Test Point − 5 −11 5

f′(x) −+−+

f(x) decr incr decr incr

Step 4: Write a conclusion. Therefore, f(x) is increasing on [−3, 0] and [3,∞) and

decreasing on (−∞,−3] and [0, 3]. (See Figure 7.2-6.)

[–8,8] by [–1,5]

Figure 7.2-6

Example 5

The derivative of a functionf is given asf′(x)=cos(x^2 ). Using a calculator, find the values

ofxon

[

−

π

2

,

π

2

]

such that f is increasing. (See Figure 7.2-7.)