Is the following statement true or false?
Suppose $f:\mathbb{R}^2\to\mathbb{R}$ is a differentiable function and $(c,d)\in\mathbb{R}^2$. If $f_{xx}(c,d)f_{yy}(c,d)<[f_{xy}(c,d)]^2$ then $f$ has a saddle point at $(c,d)$.
After some trial and error I appear to have found a counterexample with the function $f(x,y)=x^2y+xy^2$ at the point $(1,1)$, but is there a more direct or intuitive way that one can use?