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.