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

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for all <math>(x,y)</math> in <math>R</math>. | for all <math>(x,y)</math> in <math>R</math>. | ||

+ | ==First-Partials Test for Relative Extrema== | ||

+ | If <math>f</math> has a relative extremum at on an open region <math>R</math> in the xy-plane, and the first partial derivatives of <math>f</math> exist in <math>R</math>, then | ||

+ | |||

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

+ | '''Example:''' Find relative extrema of: | ||

+ | '''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 relative extrema at <math>(-2,3)</math> | ||

+ | |} | ||

− | + | ==The Second-Partials Test for Relative Extrema== | |

+ | Let <math>f</math> have continuous second partial derivatives on an open region containing <math>(a,b)</math> for which <math>f_x(a,b)=0</math> and <math>f_y(a,b)=0</math> | ||

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

## Revision as of 08:22, 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 relative extrema of:

**1)**

Solution: |
---|

Consider: , so |

and: , so |

Therefore, there is a relative extrema at |

## The Second-Partials Test for Relative Extrema

Let have continuous second partial derivatives on an open region containing for which and Then, consider

**This page were made by Tri Phan**