# 12.8.1. Increasing and Decreasing Functions

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

• A decreasing over
• B decreasing over the interval , increasing over the interval
• C decreasing over the interval , increasing over the interval
• 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 . However, the inequality can never be true, because the denominator of the fraction on the left side can never be negative, which means that the expression will always be negative, except when . Thus, the function is decreasing over .

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