# 12.8.1. Increasing and Decreasing Functions

Find the interval(s) over which the function is increasing and/or decreasing.

• A increasing over the intervals and , decreasing over the interval
• B increasing over the interval , decreasing over the intervals and
• C increasing over
• D decreasing over

### Example

Find the interval(s) over which the function is increasing and/or decreasing.

### Solution

First, find the derivative of the function as follows:

Suppose is a differentiable function over the interval . If for all , then is an increasing function of over the interval . Likewise, if for all , then is a decreasing function of over the interval . For this reason, the function is increasing when , and it is decreasing when . Now, to help determine the values of that satisfy the inequalities and , solve the equation .

This means that the zeros of the function are and . Next, graph the function .

The graph shows that the values of that satisfy the inequality are and , and the values of that satisfy the inequality are . Thus, the function is increasing over the intervals and , and it is decreasing over the interval .

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