A calculus problem about extreme value

A calculus problem about extreme value

Let $g(x,y)=0$ be a closed curve, and $g(x,y)<0$ is the interior of the
curve, and the interior is a convex set. For example, $g(x,y)=x^2+y^2-1=0$
is a closed curve(circle), and its interior is a convex set.
My question is, given a point $(x_0,y_0)$ satisfying $g(x_0,y_0)>0$, what
is the maximum of the function $f(x,y)=\nabla
g(x_0,y_0)\cdot(x,y)-g(x,y)$?
If we use the derivative, $\nabla f=\nabla g(x_0,y_0)-\nabla g(x,y)=0$,
then $(x,y)=(x_0,y_0)$ is the root of $\nabla f=0$, but how to prove the
root is the only one? and how to prove the stationary point is the maximum
point?
I think the key to solve this problem is to find out the relationship
between the convex set, $g(x,y)=0$, and its (second) derivative.