Hessian equals zero. I'm currently just working through some maxima/minima problems, but came across one that was a bit different from the 'standard' ones.
So they used the usual procedures and ended up finding that the Hessian is zero at the critical point (0,0). 
They set $x=y$, which resulted in $f(x,x)=-x^3$, which has an inflection point at the origin, which is the 2D version of the saddle point. 
I have a few questions about this.


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*How did they 'know' to set x=y, or is this a standard technique for these problems? ie: Set $x=f(y)$ and choose some convenient $f(y)$?

*In a geometric sense, what does setting $x=y$ mean? I'm having trouble visualising this. 



set x=y means evaluating the function on the line x=y. put it another way, evaluating the function along the direction $\mathbf{v}=[1,1]^T$. Since an extremum on the whole must be an extermum along any direction,. If we can find a direction along which this critical point is not an extremum, then we can assert that this point is not an extremum on the whole. 
