# 12.8.1. Increasing and Decreasing Functions

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

• A decreasing over the interval , increasing over the interval
• Bdecreasing over the interval , increasing over the interval
• Cincreasing over
• Ddecreasing over

### Example

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

### Solution

When , the function can be rewritten as , when , it can be rewritten as , and when , it can be rewritten as , or .

Therefore, the function is equivalent to the function .

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 . Since the derivative of the function's top piece is always negative, the function is always decreasing when . Now find the derivative of the function's middle piece.

Since the derivative of the function's middle piece is always 0, the function is neither increasing nor decreasing when . Next, find the derivative of the function's bottom piece.

Since the derivative of the function's bottom piece is always positive, the function is always increasing when . Thus, the function is decreasing in the interval , and it is increasing in the interval .

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