There is a version of the "second derivative test" that works for functions of two variables. The analog of df/dx for functions of two variables is the matrix

If the determinant of that matrix (at the given (x,y)) is positive then either:

1) if the point gives a minimum.

2) if the point gives a maximum.

If the determinant is negative, then the point gives a saddle point.

If the determinant is 0, the "second derivative test" won't give you an answer and, having written all of that, I now notice that is the case!

So, instead, look what happens on lines through (0, 0). On the line x= 0, f(0, y)= 0 for all y and similarly, on y= 0, f(x, 0)= 0. But on the line y= x, the graph goes up for x positive and down for x negative. That gives you the answer.