Summary of Maxima and Minima Problems

A critical point of the function [latex]f(x,y)[/latex] is any point [latex](x_{0},y_{0})[/latex] where either [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[/latex], or at least one of [latex]f_{x}(x_{0},y_{0})[/latex] and [latex]f_{y}(x_{0},y_{0})[/latex] do not exist.
A saddle point is a point [latex](x_{0},y_{0})[/latex] where [latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[/latex], but [latex](x_{0},y_{0})[/latex] 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
[latex]D=f_{xx}(x_{0},y_{0})f_{yy}(x_{0},y_{0})-\left(f_{xy}(x_{0},y_{0})\right)^{2}[/latex]

critical point of a function of two variables

the point [latex](x_{0},y_{0})[/latex] is called a critical point of [latex]f(x,y)[/latex] if one of the two following conditions holds:

[latex]f_{x}(x_{0},y_{0})=f_{y}(x_{0},y_{0})=0[/latex]
At least one of [latex]f_{x}(x_{0},y_{0})[/latex] and [latex]f_{y}(x_{0},y_{0})[/latex] do not exist

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

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