# Math 22 Increasing and Decreasing Functions

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## Definitions of Increasing and Decreasing Functions

A function is increasing on an interval when, for any two numbers and in the interval, implies

A function is decreasing on an interval when, for any two numbers and in the interval, implies

## Test for Increasing and Decreasing Functions

Let be differentiable on the interval . 1. If for all in , then is increasing on . 2. If for all in , then is decreasing on . 3. If for all in , then is constant on .

## Critical Numbers and Their Use

If is defined at , then is a critical number of when or when is undefined.

**Exercises:** Find critical numbers of

**1)**

Solution: |
---|

So, is critical number |

**2)**

Solution: |
---|

So, |

In this case, we have critical number when is undefined, which is when . So critical number is |

## Increasing and Decreasing Test

1. Find the derivative of . 2. Locate the critical numbers of and use these numbers to determine test intervals. That is, find all for which or is undefined. 3. Determine the sign of at one test value in each of the intervals. 4. Use the test for increasing and decreasing functions to decide whether is increasing or decreasing on each interval.

**Exercises:** Find the intervals of increasing and decreasing of

**1)**

Solution: |
---|

So, are critical numbers. |

Hence, the test intervals are and |

In each interval, choose a number and test on : |

So, , so is increasing on |

, so is decreasing on |

, so is increasing on |

Therefore, is Increasing: |

Decreasing on |

**This page were made by Tri Phan**