The problem is: Let $\vec{x}\in\mathbb{R}^d$ be the variable and $f(\vec{x})$ be a scalar function that is globally strictly convex in $\mathbb{R}^d$. We assume the unique optimum of $f$ to be finite(in all dimensions). Denote the optimum as $\hat{x}$. Suppose $f$ is smooth enough, can we know the sign of each dimension in $\hat{x}$ by looking at the sign of each dimension in $\nabla f(\vec{x})|_{\vec{x}=\vec{0}}$?

If yes, any proof or justification? If no, could you provide a counterexample?

Thanks much!!