What to do when the second derivative test fails? What do we do when the second derivative test fails? 
For example, I'm asked to find all the critical points of the function
$$f(x,y)=x^{2013}−y^{2013}$$
And determine the nature of the critical points. 
The critical point that I have found is at $(0,0)$, but I'm unable to determine its nature as the second derivative test fails here.

Note that $f(x,y) > 0$ whenever $x>0$ and $y<0$.  Moreover, whenever $x<0$ and $y>0$ we have $f(x,y)<0$.  Thus, as $f(0,0)=0$, we have points $(x_1,y_1)$ and $(x_2,y_2)$ in every neighborhood of $(0,0)$ such that $f(x_1,y_1) < f(0,0) < f(x_2,y_2)$.  As $f$ is defined on $\mathbb{R}^2$ and hence on the just constructed points, we see that $(0,0)$ satisfies the definition of a saddle point of $f$.
