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so I got this homework problem that I was having trouble with. The problem is: Let $f(x, y) = \sin(xy)$ defined on all of $\mathbb{R}^2$. Find the critical points of $f$ and classify them as local maxima, local minima, or saddle points.

The problem is that I calculated the following: $f_{xx} = -y^2 \sin(xy)$, $f_{yy} = -x^2 \sin(xy)$ and $f_{xy} = f_{yx} = -yx \sin(xy)$. Then, the determinant of the hessian ends up being $0$ no matter what $(x, y)$ we choose, so the 2nd derivative test is inconclusive. Am I doing something wrong here?

Thanks in advance.

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Hint: Your calculation of $f_{xy}$ is not correct: use the Product Rule.

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  • $\begingroup$ Whoops, I'm such an idiot. Thanks. $\endgroup$ – Karsh Narayan Feb 12 '14 at 20:22
  • $\begingroup$ You are welcome. A minor slip does not make one an idiot, else I am in real trouble. $\endgroup$ – André Nicolas Feb 12 '14 at 20:25

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