Classify all stationary points

$$\nabla f=(4x^3-4x,2y),$$

so the critical points are $(0,0)$, $(1,0)$, and $(-1,0)$. The Hessian is

$$H=\operatorname{diag}(12x^2-4,2)$$

At $(0,0)$ it has eigenvalues $-4,2$, so this point is a saddle. At each of $(\pm1,0)$ it is $\operatorname{diag}(8,2)$, so both are strict (indeed global) minima with value $-1$. The function is not convex on all of $\mathbb R^2$ because $\partial^2 f/\partial x^2=-4$ at the origin.