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.

Hint:


*

*Take into consideration higher-order derivatives.

*Note the parity of the first non-zero derivative. 

*What are the similarities among $x^3, x^5, x^7,\ldots$ and similarities among $x^2,x^4,x^6,\ldots$ (e.g. how the graphs would look like, and what is the parity of the first non-zero derivative)?

*You can read more about it here.


I hope this helps $\ddot\smile$
