# 12.8.1. Increasing and Decreasing Functions

Given that the function , find the intervals over which it is increasing and/or decreasing.

• Adecreasing over the interval , increasing over the interval
• Bincreasing over the interval , decreasing over the interval
• Cincreasing over the interval , decreasing over the interval
• Dincreasing over

### Example

Given that the function , find the intervals over which it is increasing and/or decreasing.

### Solution

First, find the derivative of the function's top piece, , 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 . Now use this fact to help find the values of for which the function's top piece, , is increasing.

Since the function's top piece is only defined for , it is increasing over the interval . Next, find the values of for which the function's top piece, , is decreasing.

This means the function's top piece is decreasing over the interval . Now find the derivative of the function's bottom piece, , as follows:

Since the derivative of the function's bottom piece is always greater than 0, it is always increasing over the interval in which it is defined, which is . Thus, the function is increasing over the interval , or , and it is decreasing over the interval .

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