# Difference between revisions of "Math 22 Extrema of Functions of Two Variables"

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<math>f_x(x_0,y_0)=0</math> and <math>f_y(x_0,y_0)=0</math> | <math>f_x(x_0,y_0)=0</math> and <math>f_y(x_0,y_0)=0</math> | ||

− | '''Example:''' Find relative | + | '''Example:''' Find the relative critical point of of: |

'''1)''' <math>f(x,y)=2x^2+y^2+8x-6y+20</math> | '''1)''' <math>f(x,y)=2x^2+y^2+8x-6y+20</math> | ||

Line 26: | Line 26: | ||

|and: <math>f_y(x,y)=2y-6=0</math>, so <math>y=3</math> | |and: <math>f_y(x,y)=2y-6=0</math>, so <math>y=3</math> | ||

|- | |- | ||

− | |Therefore, there is a | + | |Therefore, there is a critical point at <math>(-2,3)</math> |

|} | |} | ||

Line 39: | Line 39: | ||

4. If <math>d=0</math>, no conclusion. | 4. If <math>d=0</math>, no conclusion. | ||

+ | '''Example:''' Find the relative extrema (maximum or minimum): | ||

+ | |||

+ | '''1)''' <math>f(x,y)=2x^2+y^2+8x-6y+20</math> | ||

+ | {| class = "mw-collapsible mw-collapsed" style = "text-align:left;" | ||

+ | !Solution: | ||

+ | |- | ||

+ | |Consider: <math>f_x(x,y)=4x+8=0</math>, so <math>x=-2</math> | ||

+ | |- | ||

+ | |and: <math>f_y(x,y)=2y-6=0</math>, so <math>y=3</math> | ||

+ | |- | ||

+ | |Therefore, there is a critical point at <math>(-2,3)</math> | ||

+ | |- | ||

+ | |Now: <math>f_{xx}f(x,y)=4</math> | ||

+ | |- | ||

+ | |<math>f_{yy}f(x,y)=2</math> | ||

+ | |- | ||

+ | |and <math>f_{xy}f(x,y)=0</math> | ||

+ | |- | ||

+ | |Then, <math>d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^2=(4)(2)-0^2=8</math> | ||

+ | |- | ||

+ | |Since, <math>d>0</math> and <math>f_{xx}f(x,y)=4>0</math>, then by the second-partial test, <math>f</math> has a relative minumum at <math>(-2,3)</math> | ||

+ | |} | ||

[[Math_22| '''Return to Topics Page''']] | [[Math_22| '''Return to Topics Page''']] | ||

'''This page were made by [[Contributors|Tri Phan]]''' | '''This page were made by [[Contributors|Tri Phan]]''' |

## Latest revision as of 08:32, 18 August 2020

## 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**