How would I do $f(x, y) = 5x^2 + 6y^2$ on circle $x^2+y^2 = 1$?
I understand that we first solve for the critical points getting $f(0, 0) = 0$ which is the global minimum. But after that how do we get the points at the boundary. I know $x = \arccos(\theta)$ and $y = \arcsin(\theta)$ but after putting those into our $f(x,y)$ I get confused when taking the derivative of that new function. Can anyone help or give a hint?
On the boundary, you solve the system$$\left\{\begin{array}{l}10x=2\lambda x\\12y=2\lambda y\\x^2+y^2=1\end{array}\right.$$It's solutions are $(x,y)=(\pm1,0)$ and $(x,y)=(0,\pm1)$. So, see what's the value of your function at these four points.