# Math 22 Extrema of Functions of Two Variables

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## Relative Extrema of a Function of Two Variables

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

${\displaystyle f(x,y)\leq f(x_{0},y_{0})}$

for all ${\displaystyle (x,y)}$ in ${\displaystyle R}$.

 The function ${\displaystyle f}$ has a relative minimum at ${\displaystyle (x_{0},y_{0})}$ when there is a circular region  centered at ${\displaystyle (x_{0},y_{0})}$ such that

${\displaystyle f(x,y)\geq f(x_{0},y_{0})}$

for all ${\displaystyle (x,y)}$ in ${\displaystyle R}$.


## First-Partials Test for Relative Extrema

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

${\displaystyle f_{x}(x_{0},y_{0})=0}$ and ${\displaystyle f_{y}(x_{0},y_{0})=0}$


Example: Find the relative critical point of of:

1) ${\displaystyle f(x,y)=2x^{2}+y^{2}+8x-6y+20}$

Solution:
Consider: ${\displaystyle f_{x}(x,y)=4x+8=0}$, so ${\displaystyle x=-2}$
and: ${\displaystyle f_{y}(x,y)=2y-6=0}$, so ${\displaystyle y=3}$
Therefore, there is a critical point at ${\displaystyle (-2,3)}$

## The Second-Partials Test for Relative Extrema

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

Then:
1. If ${\displaystyle d>0}$ and ${\displaystyle f_{xx}(a,b)>0}$, then ${\displaystyle f}$ has a relative minimum at ${\displaystyle (a,b)}$.
2. If ${\displaystyle d>0}$ and ${\displaystyle f_{xx}(a,b)<0}$, then ${\displaystyle f}$ has a relative maximum at ${\displaystyle (a,b)}$.
3. If ${\displaystyle d<0}$, then ${\displaystyle (a,b,f(a,b))}$ is a saddle point.
4. If ${\displaystyle d=0}$, no conclusion.


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

1) ${\displaystyle f(x,y)=2x^{2}+y^{2}+8x-6y+20}$

Solution:
Consider: ${\displaystyle f_{x}(x,y)=4x+8=0}$, so ${\displaystyle x=-2}$
and: ${\displaystyle f_{y}(x,y)=2y-6=0}$, so ${\displaystyle y=3}$
Therefore, there is a critical point at ${\displaystyle (-2,3)}$
Now: ${\displaystyle f_{xx}f(x,y)=4}$
${\displaystyle f_{yy}f(x,y)=2}$
and ${\displaystyle f_{xy}f(x,y)=0}$
Then, ${\displaystyle d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}=(4)(2)-0^{2}=8}$
Since, ${\displaystyle d>0}$ and ${\displaystyle f_{xx}f(x,y)=4>0}$, then by the second-partial test, ${\displaystyle f}$ has a relative minumum at ${\displaystyle (-2,3)}$