Upper bound of a function

Question

I need to find the upper bound of the function $f(x,y)=x(y-x-1)e^{-y}$ on $A=\{(x,y)\mid 0\le x\le y\}$.

So $f_{x}=e^{-y}(y-2x-1)$ and $f_{y}=xe^{-y}(x-y+2)$ and critical points are: $(0,1)$ and $(1,3)$ but here come my doubts. $A$ is not compact so I can't use Weierstrass' theorem? So how can I examine it?

Answer 1

Hint: Show that $f$ is bounded above on $\{y=c\}\cap A$ by very small positive numbers as $c\to \infty.$ This allows you to reduce to a compact subset of $A.$