# Absolute extrema of a multivariable function

Consider the function:

$$f(x,y) = x^3 + y^3 + 3xy,\ x,y \in \mathbb{R}$$

Setting the gradient to zero we find that there are two candidates: $\alpha = (-1,-1)$ and $\beta = (0,0)$, looking at the hessian matrix we find that $\alpha$ is a maximum and $\beta$ a saddle point.

but $\alpha$ is actually not only a local but an absolute maximum, how do we determine that in general and specifically in this case?

How would it be the global maximum? Say, $\lim_{x\rightarrow\infty}f(x,1)=\lim_{x\rightarrow\infty}(x^{3}+3x+1)=\infty$. So for any $M>f(\alpha)=f(-1,-1)$, there is an $(x_{M},1)$ such that $f(x_{M},1)>M>f(\alpha)$, so $f(\alpha)$ is not a global maximum.
I don't think it is an "absolute" maximum value since for $$x,y\to +\infty \implies f(x,y) \to +\infty$$