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 the relative critical point of 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 critical point 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 $d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}$ Then:
1. If $d>0$ and $f_{xx}(a,b)>0$ , then $f$ has a relative minimum at $(a,b)$ .
2. If $d>0$ and $f_{xx}(a,b)<0$ , then $f$ has a relative maximum at $(a,b)$ .
3. If $d<0$ , then $(a,b,f(a,b))$ is a saddle point.
4. If $d=0$ , no conclusion.

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

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 critical point at $(-2,3)$ Now: $f_{xx}f(x,y)=4$ $f_{yy}f(x,y)=2$ and $f_{xy}f(x,y)=0$ Then, $d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}=(4)(2)-0^{2}=8$ Since, $d>0$ and $f_{xx}f(x,y)=4>0$ , then by the second-partial test, $f$ has a relative minumum at $(-2,3)$ 