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The second derivative test for functions of two variables says to first find critical points. For each critical point one finds $$ D = f_{xx}f_{yy} - f_{xy}^2 $$ If $D>0$, the sign of $f_{xx}$ says something about whether the point is a local maximum or local minimum.

My question is: Why do we use $f_{xx}$? Could we use $f_{yy}$ instead?

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  • $\begingroup$ If $D>0$ they have the same sign, so yes. $\endgroup$ – Randall Oct 5 '18 at 14:54
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    $\begingroup$ +1 for curiosity & symmetry observation! $\endgroup$ – lisyarus Oct 5 '18 at 15:18
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You are assuming that $D >0$. This says that $$ f_{xx}f_{yy} > f_{xy}^2 \geq 0. $$ Hence either both $f_{xx}, f_{yy}$ are positive together or negative together. Since they have the same sign, the test works either way.

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