Question

How do you optimize `f(x,y)=2x^2+3y^2-4x-5` subject to `x^2+y^2=81`?

Answer 1

Maximum is 242, at `(-2, +-sqrt77 )`
There is no minimum.

Explanation:

Using `x^2+y^2=81`,
f(x, y) = g(x) = `238-x^2-4 x-5`
`d/dxg(x)=-2x-4 = 0`, when `x=-2`.
`d^2/dx^2g(x)--2<0`.
When `x=-2,y=+-sqrt 77`.
So, at `(-2,+-sqrt77)`, f(x,y) attains its maximum value.
There is no other extreme value.