Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex function differentiable on $\mathbb{R}^n$, and $x^{\star} \in \mathbb{R}^n$. Show that if $\nabla f(x^{\star}) = 0$ then $f$ admits a global minimum at $x^{\star}$.
Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ be a convex function differentiable on $\mathbb{R}^n$, and $x^{\star} \in \mathbb{R}^n$. Show that if $\nabla f(x^{\star}) = 0$ then $f$ admits a global minimum at $x^{\star}$.