relative maxima and minima in a closed region

I'm working on a problem and it requires the relative maxima and minima of a closed region, I understand how to find the rel(max/min) of a function f(x,y) without the closed region(using fx, fy,fxy,fxx,fxy). Does anyone know how to do it with a closed region(ex circle x^2 + y^2 <= 1). Does it mean that all points outside of the circle I do not need to examine?

Re: relative maxima and minima in a closed region

One trick I've seen used for a circular boundary is to parametrize the boundary so that you have one variable with which to work.

In order to examine on the boudary of the region, we represent the circle by means of the parametric equations . Thus, on the boundary we can write as a function of a single variable :