relative maxima and minima in a closed region

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