Consider a function like $$f(x,y)=\frac{1}{1+\sqrt[3]{x^2}+\sqrt[3]{y^2}}$$
It is obvious that the maximum of the function is at $x=y=0$.
However if t we try to apply the second derivative test, we will have to evaluate the Hessian at a point where it is not defined.
It will contains elements such as $$\frac{2(5x^\frac{2}{3}+y^\frac{2}{3}+1)}{9x^\frac{4}{3}(x^\frac{2}{3}+y^\frac{2}{3}+1)^3} $$
The element is not defined at zero, how do we overcome this, do we take the limit of the Hessian matrix at $(0, 0)$ or how exactly?
I am not asking for a solution, rather to a way to just overcome the singularity in the Hessian.
The thing is the point $(0,0)$ is not a critical point to be tested for Hessian. In fact there is no critical point: $f_x\ne 0, f_y\ne 0$. The border point $(0,0)$ provides the global max, for which Hessian is not required, unless KKT.