Question

We want to find the maximum and minimum values of f(x, y) = 10x2 + 11y on the disk D: x2 + y2 < 1.

Answer 1

If you're just focusing on the boundary, it's the circle $x^2+y^2=1$. Parameterize it by taking $x=\cos t$ and $y=\sin t$, with $0\le t\le2\pi$. Then $f(x,y)$ reduces to a univariate function $g(t)$:

$f(x,y)=10x^2+11y\iff g(t)=f(\cos t,\sin t)=10\cos^2t+11\sin t$

Then you can find any extrema on the boundary by checking the critical points of $g$.