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 $(-2,3)$

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