Question

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

Now focus on the boundary of D, and solve for y2. Restricting f(x,y) to this boundary, we can express f(x,y) as a function of a single variable x. What is this function and its closed interval domain?

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`.