Suppose we have a smooth differentiable planar curve given as $F(x,y)=0$. Suppose that the centroid of the curve $F=0$ is $F(x_0,y_0)$.
My question is:- $$\nabla F(x_0,y_0)=0$$ That is, does the gradient of $F$ becomes zero at $(x_0,y_0)$?
Now some examples supporting this statement are circles and ellipses.
I would be grateful if anyone proves the statement or gives a counter example. Also I welcome anyone who would like to generalise this statement in any direction.