Show that the maximumof $f$ isglobal

Question

I want to show that $f(x,y)=xye^{-x-y}$ has a global maximum in $(1,1)$.

So we have to show that $f(x,y)\leq f(1,1)$, or not?

We have that $$xye^{-x-y}=\frac{xy}{e^{x+y}}\leq xy$$ But how can we continue? I don't really have an idea. Could you give me a hint?

Answer 1

First of all your claim is not true if the domain includes $x<0,y<0$.

Hint:

For $x,y>0$ just write

$$xye^{-(x+y)}=\left(xe^{-x}\right)\left(ye^{-y}\right)$$

and note that $te^{-t}>0$ attains for $t>0$ its maximum $\frac 1e$ at $t=1$.