I have to show that $(1,1)$ a strict global maximum for $(x,y)\rightarrow xye^{-x-y}$ with $x > 0$ and $y > 0$.
First I calculated the derivatives: $$\frac{∂f}{∂x}=y(x-1)(-e^{-x-y}) $$$$\frac{∂f}{∂y}=x(y-1)(-e^{-x-y})$$
Therefore I know that $(1,1)$ and $(0,0)$ are the critical points. With the second derivative test I get that $$\frac{∂^2f}{∂y²}(1,1)\leq0$$ and therefore (1,1) is a local maximum. Now how do I show, that $(1,1)$ is also a global and a strict maximum in a two variable function?