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

Relative Extrema of a Function of Two Variables

Let $f$ be a function defined on a region containing $(x_{0},y_{0})$ . The function $f$ has a relative maximum at $(x_{0},y_{0})$ when there is a circular region  centered at $(x_{0},y_{0})$ such that

$f(x,y)\leq f(x_{0},y_{0})$ for all $(x,y)$ in $R$ .
The function $f$ has a relative minimum at $(x_{0},y_{0})$ when there is a circular region  centered at $(x_{0},y_{0})$ such that

$f(x,y)\geq f(x_{0},y_{0})$ for all $(x,y)$ in $R$ .

First-Partials Test for Relative Extrema

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

$f_{x}(x_{0},y_{0})=0$ and $f_{y}(x_{0},y_{0})=0$ Example: Find relative extrema of:

1) $f(x,y)=2x^{2}+y^{2}+8x-6y+20$ Solution:
Consider: $f_{x}(x,y)=4x+8=0$ , so $x=-2$ and: $f_{y}(x,y)=2y-6=0$ , so $y=3$ Therefore, there is a relative extrema at $(-2,3)$ The Second-Partials Test for Relative Extrema

Let $f$ have continuous second partial derivatives on an open region containing $(a,b)$ for which $f_{x}(a,b)=0$ and $f_{y}(a,b)=0$ Then, consider $d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}$ 