Suppose we have a continuous and differentiable function $f(x,y)$ defined on $x^{2}+y^{2} \leq 1$. Let's say this function has a maxima at some point $r$ inside this region, that is, in $x^2 + y^2<1$.
My question: If this function has no extrema (that is it's derivatives don't go to zero anywhere except at $r$) except the one specified above in this region ,is it possible for the function to have a greater value in this region than the maxima we had above?.