# Math 22 Extrema of Functions of Two Variables

## 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 ${\displaystyle R}$.