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.

(I was having a bit of indecision as to whether this merited being an answer, or just a comment, since it is related to the answers already posted.)
If we note the "cross-sections" in the $ \ yz- \ (x = 0) \ $ and $ \ xz- \ (y=0) \ $ planes, we note that the function $ \ f(x,y) \ $ reduces to $ \ f(0,y) \ = \ -y^{2013} \ $ and $ \ f(x,0) \ = \ x^{2013} \ $ .  Both of these "reduced" functions have odd symmetry about the $ \ z-$ axis.
For a function of two variables, a point cannot be a local extremum, but will be a saddle point, if either one or both of these "cross-sectional" functions is odd about the axis through that point.  If both of the functions are even, both must have the same direction of concavity about that point in order for it to be a local extremum.  If the two functions have opposite concavities, the point will be a saddle point.  (This is related to the concept of Gaussian curvature -- in that view, a local extremum has $ \ K \ > \ 0 \ $ .)
So at the origin, we will find that $ \ x^{odd} \ \pm \ y^{odd} \ $ will produce a saddle point there, $ \ \pm \ (x^{even} \ + \ y^{even}) \ $ has a local extremum there, and will be a saddle point for $ \ \pm \ (x^{even} \ - \ y^{even}) \ $ .  This can be extended to more complicated symmetrical functions and to functions with more variables.  (A local extremum requires more care to construct in larger numbers of dimensions, since the directions of concavity must agree in all of those dimensions.)
[EDIT -- In addition, it might be noted that we will also have a saddle point at the origin if the parities of the "cross-sectional" functions differ, as with, for instance $ \ x^{odd} \ \pm \ y^{even} \ $ . ]
The "second-derivative test", like a lot of mathematical "tools", has a limit to its utility.  We must expect to do some extra investigation on critical points when we deal with power functions with exponents greater than two.
