Hey guys. I am having some trouble visualizing one aspect of the Second derivative test in the 2 variable case (related to #3 below). Essentially, what does the curve look like when [itex]f_{xx}f_{yy} > 0[/itex], BUT [itex]f_{xx}f_{yy} < [f_{xy}]^{2}[/itex]?(adsbygoogle = window.adsbygoogle || []).push({});

To be more detailed, if the function is f(x,y), H(x,y) is the Hessian matrix of f and D is the determinant of H, where [itex]D = Det(H(x,y)) = f_{xx}f_{yy} - [f_{xy}]^{2} [/itex]

1) If D(a, b) > 0, and [itex]f_(xx)(a,b) > 0[/itex] => local minimum

2) If D(a, b) > 0, and [itex]f_(xx)(a,b) < 0[/itex] => local maximum

3) If D(a, b) < 0 => saddle point

I can totally see why [itex]f_{xx}[/itex] and [itex]f_{yy}[/itex] must have the same sign for there to be a max or a minimum - but I DON'T see why the product has to be "greater" than the square of [itex]f_{xy}[/itex] (as opposed to just 0) to have a max or min.

Thanks guys. Much appreciated.

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# Visualizing second derivative test - Hessian

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