5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 17:21

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

Solution:(See Figure 8.2-5.)

f′

x

f decr.

1

––

(^256)

• incr. decr.
  Figure 8.2-5
  Thus,fis decreasing on [1, 2] and [5, 6] and increasing on [2, 5].
  Example 4
  Find the open intervals on whichf(x)=(x^2 −9)^2 /^3 is increasing or decreasing.
  Step 1: Find the critical numbers off.
  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 the intervals.

• 3 03
  The 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 8.2-6.)

[–8, 8] by [–1, 5]

Figure 8.2-6