Calculus AB and Calculus BC

CHAPTER 4 Applications of Differential Calculus

C. INCREASING AND DECREASING FUNCTIONS

CASE I. FUNCTIONS WITH CONTINUOUS DERIVATIVES.

A function y = f (x) is said to be Image on an interval if for all a and b in the interval such that a < b, Image To find intervals over which f (x) Image that is, over which the curve Image analyze the signs of the derivative to determine where Image

EXAMPLE 10

For what values of x is f (x) = x4 − 4x3, increasing? decreasing?

SOLUTION: f (x) = 4x3 − 12x2 = 4x2 (x − 3).

With critical values at x = 0 and x = 3, we analyze the signs of f in three intervals:

Image

The derivative changes sign only at x = 3. Thus,

if x < 3 f (x) ≤ 0 and f is decreasing;

if x > 3 f (x) > 0 and f is increasing.

Note that f is decreasing at x = 0 even though f (0) = 0. (See Figure N4–5.)

CASE II. FUNCTIONS WHOSE DERIVATIVES HAVE DISCONTINUITIES.

Here we proceed as in Case I, but also consider intervals bounded by any points of discontinuity of f or f .

EXAMPLE 11

For what values of x is Image increasing? decreasing?

SOLUTION: Image

We note that neither f nor f is defined at x = −1; furthermore, f (x) never equals zero. We need therefore examine only the signs of f (x) when x < −1 and when x > −1.

When x < −1, f (x) < 0; when x > −1, f (x) < 0. Therefore, f decreases on both intervals. The curve is a hyperbola whose center is at the point (−1,0).