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I know if $f_{xx}*f_{yy}-f_{xy}^2 < 0$ it's a saddle point, $= 0$ is inconclusive, and $> 0$ is an extrema point. And if it's an extrema you use whether $f_{xx}$ is positive or negative to determine if it's a max or min. A few times when mis classifying saddle points I've tried to plug them into $f_{xx}$ and wound up getting $f_{xx}=0$ as a result. Is this something that always is? Can you tell it is a saddle point just off $f_{xx}$ Aside from having found an incorrect critical point that isn't really a critical point, I can't think of an exception. But considering I have a very heavily weighted test upcoming, I'd rather be sure I'm not missing anything.

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If $f_{xx} = 0$, then $f_{xx} f_{yy} - f_{xy}^2 = - f_{xy}^2$. This is never positive, would be $0$ if $f_{xy} = 0$, otherwise is negative.

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