The task is to prove that $f(x,y)=frac{x^2-4xy+2y^2}{e^{|x|+|y|}}$ is limited and to find its biggest and lowest values. The only method of solving such a problem, that I have studied, is using Weierstrass’ theorem for multiple variables:

If f(x,y) is differentiable inside a compact space C (compact space = closed limited set) and continuous on the contour(border) of the space, then f reaches its biggest and lowest values inside C.

This is an approximate translation of the theorem. The problem however is this: there is no explicitly defined compact space for the function, as it is defined everywhere in $R^2$.

I tried evaluating $frac{x^2-4xy+2y^2}{e^{|x|+|y|}}$because I have the feeling it’s bigger than zero but I couldn’t find any boundaries for the expression. The only limit I proved is $limlimits_{(x,y)to(infty, infty)}{frac{x^2-4xy+2y^2}{e^{|x|+|y|}}}=0$ but that doesn’t really help me in any way. If I manage to find a compact space for the function, I would be able to procede with the problem.

If you know any other method of proving the problem, it also helps.

Any help would be appreciated.