Region $D$: where $D = \{(x, y) \mid x \geq 0, y \geq 0, x^2+y^2 \leq 9\}$
I am able to find the critical points of the equation, however when attempting to find the critical points on the boundary of the circle I get lost. Any guidance would be really appreciated, thanks guys.
Here is the problem and my work:
Since you have a compact domain, the function has to attain its maximun and minimun in it, so you also have to check points in the border. To do that, you could parametrise the circle which is the boundary of your domain. This is going to be a curve in $\mathbb{R}^2$, labeled as $ g : \mathbb{R} \longrightarrow \mathbb{R}^2$ where $g(t)=(3\cos t, 3\sin t)$ and then you just compound with your function $f$, so you get a map $f \circ g : \mathbb{R} \longrightarrow \mathbb{R}$ and you know how to minimise or maximise a one real variable function, by looking at its derivatives.