1. ## Finding absolute extremaa

Let $\displaystyle f(x,y)=x^{3}-3x-y^{3}+12y$. Find the absolute maximum and absolute minimum of f on the region $\displaystyle D=\{(x,y)|x^{2}+y^{2}\leq 1\}$

So I tried to find critical points, and the only ones are $\displaystyle (\pm 1,\pm 2)$, which are not in the region $\displaystyle D$, so now I have to find the extrema on the boundary. This is where I am having much trouble. I tried doing both $\displaystyle f(x,\sqrt{1-x^{2}})$ and $\displaystyle f(\sqrt{1-y^{2}},y)$, and I end with pretty nasty equations that seem impossible (or at least extremely difficult) if I set their derivatives equal to 0. Any tips/work shown would be greatly appreciated.

2. No takers? I tried using the substitution $\displaystyle x=cos(t),y=sin(t)$ but it was still an extremely convoluted equation.

3. So, I redid trig substitution and eventually ended up with $\displaystyle g'(\theta)=3(sin^{3}\theta + cos^{3}\theta +3cos\theta)$. I'm stuck on how to solve for $\displaystyle g'(\theta)=0$

4. Originally Posted by Pinkk

So, I redid trig substitution and eventually ended up with $\displaystyle g'(\theta)=3(sin^{3}\theta + cos^{3}\theta +3cos\theta)$. I'm stuck on how to solve for $\displaystyle g'(\theta)=0$
if you let $\displaystyle \tan \theta = t,$ then $\displaystyle g'(\theta)=0$ will give you $\displaystyle t^3 + 3t^2 + 4=0,$ which has 3 real distinct solutions none of which are nice numbers!