Here is a question that I have been working on but having trouble with.
Let $f(x)=e^{-|x|^2}$, where $x \in \mathbb{R}^n$ and $|x|$ the usual euclidean norm of $x$.
- Prove that for every $\epsilon >0$ there is a positive number $M$ such that $g(x,y):=f(x)g(y)|x-y|^2 < \epsilon$ whenever $|x|^2+|y|^2 >M$. I showed this Using the fact that $e^{-|x|^2}$ goes to zero as norm of $x$ goes to infinity. But I'm having trouble with the 2nd and 3rd part of the question.
- Show that $S:=\sup_{x,y\in \mathbb{R}^n}f(x)f(y)|x-y|^2$ is attained at some point in $\mathbb{R}^n \times \mathbb{R}^n$.
- Determine the value of S.