Summary of Maxima and Minima Problems

A critical point of the function $f(x,y)$ is any point $(x_{0},y_{0})$ where either $f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0$ , or at least one of $f_{x}(x_{0},y_{0})$ and $f_{y}(x_{0},y_{0})$ do not exist.
A saddle point is a point $(x_{0},y_{0})$ where $f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0$ , but $(x_{0},y_{0})$ is neither a maximum nor a minimum at that point.
To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test.

Key Equations

Discriminant
$D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\left(f_{xy}(x_{0},y_{0})\right)^{2}$

critical point of a function of two variables

the point

(x_{0}, y_{0})

$(x_{0},y_{0})$ is called a critical point of

f (x, y)

$f(x,y)$ if one of the two following conditions holds:

discriminant: the discriminant of the function $f(x,y)$ is given by the formula $D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\left(f_{xy}(x_{0},y_{0})\right)^{2}$

saddle point: given the function $z=f(x,y)$ the point $(x_{0},y_{0},f(x_{0},y_{0}))$ is a saddle point if both $f_{x}(x_{0},y_{0})=0$ and $f_{y}(x_{0},y_{0})=0$ , but $f$ does not have a local extremum at $(x_{0},y_{0})$