I just need a clarification regarding argmin and minima and critical points. As I understand it, $\text{argmin}_{x\in\mathbb{Z}}(4x^{2}-x^{4}):=\{-2,0,2\}$, but the minima of $(4x^{2}-x^{4}):=\{0\}$ and the corresponding critical point is $0$. Is this correct?
If yes, am I right in assuming that argmin does not always have to be the critical point (since there is no inflection)? Or, rather, non-critical points can also belong to the argmin set?