Essential Concepts
- 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]
Glossary
- 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]
