So I have to find the maximum and minimum value that the function $~xe^{x^2+y^2}~$ can take on: $$ D = \bigl\{(x,y) :\, 9 \leq x^2 + y^2 \le 16,~ y \geq 0\bigr\} $$
I've converted the Cartesian function into the polar function $~r\sin(\theta)e^{r^2}~$ but I'm not sure as to how I can find the max and min values from there on.
Looking at the function. It will be maximum when $x$ and $x^2+y^2$ are maximum. Given the constraints, $x=4$ and $x^2+y^2=16$ are the maximum values. Similarly, minimum would be when x=0 and it is independent of y.
In cartesian domain, maximum is when $\theta=\pi/2 $ and r=4 and minima when $\theta =0 $