# Math 22 Extrema of Functions of Two Variables

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## Relative Extrema of a Function of Two Variables

Let be a function defined on a region containing . The function has a relative maximum at when there is a circular region centered at such that for all in .

The function has a relative minimum at when there is a circular region centered at such that for all in .

## First-Partials Test for Relative Extrema

If has a relative extremum at on an open region in the xy-plane, and the first partial derivatives of exist in , then and

**Example:** Find the relative critical point of of:

**1)**

Solution: |
---|

Consider: , so |

and: , so |

Therefore, there is a critical point at |

## The Second-Partials Test for Relative Extrema

Let have continuous second partial derivatives on an open region containing for which and Then, consider Then: 1. If and , then has a relative minimum at . 2. If and , then has a relative maximum at . 3. If , then is a saddle point. 4. If , no conclusion.

**Example:** Find the relative extrema (maximum or minimum):

**1)**

Solution: |
---|

Consider: , so |

and: , so |

Therefore, there is a critical point at |

Now: |

and |

Then, |

Since, and , then by the second-partial test, has a relative minumum at |

**This page were made by Tri Phan**