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Test for Increasing and Decreasing Functions for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at:  Graphs of Functions and Derivatives Practice Problems for AP Calculus

Let f be a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b).

1. If f '(x) > 0 on (a, b), then f is increasing on [a, b].
2. If f '(x) < 0 on (a, b), then f is decreasing on [a, b].
3. If f '(x) = 0 on (a, b), then f is constant on [a, b].

Definition: Let f be a function defined at a number c. Then c is a critical number of f if either f '(c) = 0 or f '(c) does not exist. (See Figure 7.2-1.)

Example 1

Find the critical numbers of f(x) = 4x3 + 2x2.

To find the critical numbers of f(x), you have to determine where f '(x) = 0 and where f '(x) does not exist. Note f '(x) = 12x2 + 4x, and f '(x) is defined for all real numbers. Let f '(x) = 0 and thus 12x2 + 4x = 0 which implies 4x(3x +1) = 0 x = – 1/3 or x = 0. Therefore the critical numbers of f are 0 and –1/3. (See Figure 7.2-2.)

Example 2

Find the critical numbers of

Note that f '() is undefined at x = 3 and that f '(x) ≠ 0. Therefore, 3 is the only critical number of f. (See Figure 7.2-3.)

Example 3

The graph of f ' on (1, 6) is shown in Figure 7.2-4. Find the intervals on which f is increasing or decreasing.

See Figure 7.2-5.

Thus, f is decreasing on [1, 2] and [5, 6] and increasing on [2, 5].

Example 4

Find the open intervals on which is increasing or decreasing.

Step 1: Find the critical numbers of f.

Set f '(x) = 0 4x = 0 or x = 0.
Since f '(x) is a rational function, f '(x) is undefined at values where the denominator is 0. Thus, set x2 – 9 = 0 x = 3 or x = – 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.

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.)

Example 5

The derivative of a function f is given as f '(x) = cos(x2). Using a calculator, find the values of x on such that f is increasing. (See Figure 7.2-7.)

Using the [Zero] function of the calculator, you obtain x =1.25331 is a zero of f ' on Since f '(x)= cos(x2) is an even function, x = –1.25331 is also a zero on

(See Figure 7.2-8.)

Thus f is increasing on [–1.2533, 1.2533].

Practice problems for these concepts can be found at:

Graphs of Functions and Derivatives Practice Problems for AP Calculus

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