Exponential function, domain of definition

Question

I have the function $\displaystyle f(x,y)=x^2e^{-x^2-y^2}$ with the domain of definition = $\{(x,y) \mid x^2+y^2=2\}$

The task is to decide $f$'s maximum and minimum value and the range. How do I get there?

Answer 1

For $(x,y) \in A=\{(x,y) |x^2+y^2=2\}$ you have:

$$f(x,y)=x^2e^{-x^2-y^2}=x^2e^{-(x^2+y^2)}=x^2e^{-2}$$

so $f(x,y)$ is maximal when $x^2$ is maximal, so for $(\sqrt{2},0)$ and $(-\sqrt{2},0)$.