Critical Point Classification: Inconclusive Hessian

[Gekko]

What is the general approach to take when the Hessian is inconclusive when classifying critical points? ie the determinant = 0?

 

[Raskolnikov]

If det(H) = 0 and H has both positive and negative eigenvalues at x, then x is a saddle point for the function.

If not...then classifying degenerate critical points [det(H) = 0] becomes quite difficult from what I know. Thom's Splitting Lemma might work. It's sort of a parametrized version of the Morse lemma.
http://en.wikipedia.org/wiki/Splitting_lemma_(functions )

In general, I think it's safe to say that degenerate critical points are annoying haha.