Suppose [math]\left(a,b\right)[/math] is a point in the domain of [math]f\left(x,y\right)[/math] where both [math]f_x\left(a,b\right)=0[/math] and [math]f_y\left(a,b\right)=0[/math]. This means that [math]\left(a,b\right)[/math] is a [i]critical point[/i] of [math]f[/math]. Furthermore,[br][list=1][*]If [math]f_{xx}f_{yy}-f_{xy}^2>0[/math] and [math]f_{xx}<0[/math] at [math]\left(a,b\right)[/math], then [math]f[/math] has a [i]local maximum[/i] at [math]\left(a,b\right)[/math].[/*][*]If [math]f_{xx}f_{yy}-f_{xy}^2>0[/math] and [math]f_{xx}>0[/math] at [math]\left(a,b\right)[/math], then [math]f[/math] has a [i]local minimum[/i] at [math]\left(a,b\right)[/math].[/*][*]If [math]f_{xx}f_{yy}-f_{xy}^2<0[/math] at [math]\left(a,b\right)[/math], then [math]f[/math] has a [i]saddle point[/i] at [math]\left(a,b\right)[/math].[/*][*]If [math]f_{xx}f_{yy}-f_{xy}^2=0[/math] at [math]\left(a,b\right)[/math], then [i]the test is inconclusive[/i].[/*][/list]In the interactive figure, you can enter a function [math]f\left(x,y\right)[/math], then rotate the view to see where its extreme values lie. Several examples are given, or you can define your own function. Drag the red point to set [math]\left(a,b\right)[/math], and consider the relationship between the derivatives of [math]f[/math] at [math]\left(a,b\right)[/math] and the shape of the [math]f[/math]-graph.

[i]Developed for use with Thomas' Calculus and [url=https://www.pearson.com/en-us/subject-catalog/p/interactive-calculus-early-transcendentals-single-variable/P200000009666]Interactive Calculus[/url], published by Pearson.[/i]