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) Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (-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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f_{x}(a,b)=0}
 and $f_{y}(a,b)=0$
 Then, consider Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d=f_{xx}(a,b)f_{yy}(a,b)-[f_{xy}(a,b)]^{2}}

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