Where the Gradient Vanishes

A surface is given by

$f(x, y) = x^3 - 3xy + y^3$

The gradient

$\nabla f = \left(\frac{\partial f}{\partial x},\ \frac{\partial f}{\partial y}\right)$

is the zero vector at a point only when both partial derivatives vanish there at the same time — the ground is momentarily flat in both axis directions at once.

Exactly one of the points below is such a point. Which one?