# Find critical points of sin(x*y)

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?

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