What should I do if the second partial derivative test is inconclusive/0?

Question

In multivariable calculus, the second partial derivative test is used to determine whether a point of interest is a saddle point or an exetremum. First, I will not state the test here, Khan Academy has a great multivariable calculus series already. Now, I wondered what happens if the test is inconclusive (i. e. it equals 0). I searched for that, and I didn't find the answer. And please don't answer "Ah, just use a graph.". I want a conclusive answer.

Answer 1

Start with the Hessian matrix. Diagonalizing it gives you a set of principal directions. "Second derivative test fails" means that in at least one of the directions, the second derivative is $0$. So looking at higher derivatives in that direction tells you the true state of that direction. If you have local minimum in at least one direction and local maximum in at least one direction, then this is a saddle point. Local minimum in all directions means this is a local minimum. Local maximum in all directions means this is a local maximum. A direction of neither (think of $x^3$) tells us this is neither local maximum nor local minimum.

We may still be left with peculiar cases. Some direction where all derivatives are zero but the function is not identically zero. (Think of $\exp(-1/x^2)$).

Another situation could be where some derivative you need to look at fails to exist (think of $|x|$).