## 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 on an interval if for all *a* and *b* in the interval such that *a* < *b*, To find intervals over which *f* (*x*) that is, over which the curve analyze the signs of the derivative to determine where

**EXAMPLE 10**

For what values of *x* is *f* (*x*) = *x*^{4} − 4*x*^{3}, increasing? decreasing?

**SOLUTION:** *f* *′*(*x*) = 4*x*^{3} − 12*x*^{2} = 4*x*^{2} (*x* − 3).

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

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 increasing? decreasing?

**SOLUTION:**

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).