# Second derivative test using $f_{yy}$ instead of $f_{xx}$?

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?

• If $D>0$ they have the same sign, so yes. – Randall Oct 5 '18 at 14:54
• +1 for curiosity & symmetry observation! – lisyarus Oct 5 '18 at 15:18

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.