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 $\displaystyle f$ on the boudary of the region, we represent the circle $\displaystyle x^2+y^2=1$ by means of the parametric equations $\displaystyle x=\cos(t),\,y=\sin(t),\,0\le t\le2\pi$. Thus, on the boundary we can write $\displaystyle f$ as a function of a single variable $\displaystyle t$:

$\displaystyle f(t)=f(\cos(t),\sin(t))$