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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.

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  • $\begingroup$ For a fixed value of $\theta$, your function is increasing in $r$, and your region is $3\leq r\leq 4$ and $-\pi/2\leq \theta \leq \pi/2$. Because the function is a product of function of $r$ and a function of $\theta$, and because your bounds are a "rectangle" in the $(r,\theta)$-plane, you can essentially analyze the behavior in each coordinate separately. $\endgroup$ – Aaron Dec 8 '15 at 15:38
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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 $

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