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.