To evaluate the minima, maxima, and saddle points of a real function of 2 variables, we use the second derivative test after evaluating the critical points to identify the type of extrema they are. Recall that given a function $f(x,y)$ and the Hessian matrix function $H(f)$, the second derivative test tells us
If $det(H)(x_0,y_0)> 0$, and $f_{xx}(x_0, y_0) < 0$, then $(x_0,y_0)$ is a local maximum.
If $det(H)(x_0,y_0)> 0$, and $f_{xx}(x_0, y_0) > 0$, then $(x_0,y_0)$ is a local minimum.
If $det(H)(x_0,y_0)< 0$, then $(x_0,y_0)$ is a saddle point.
If $det(H)(x_0,y_0)=0$, then the test is inconclusive.
What I am most interested in is the last of these bullets. What is there left to do besides look at the graph in case the second derivative test yields inconclusive results? For single-variable functions, you would use a higher order derivative. However, a higher order derivative would have $2^o$ derivatives (for order $o$), so what calculation would be necessary to identify the type of critical point in this case?
