Are the stationary points of a strongly convex function unique in each dimension?

Question

Consider a strongly convex function $~f: \mathbb{R}^n \rightarrow \mathbb{R^+}~$ with a unique minimum at the point $x^* \in \mathbb{R}^n$.

I am wondering: if I have another point $y \in \mathbb{R}^n$ such that $\nabla{f}(y)^T e_i = 0$ for some $i = 1 \ldots n$, then is it true that $y_i = x_i^*$.

If so, I was hoping that someone could outline a proof. If not, can someone provide a counter example that is easy to visualize?

Answer 1

Let $f(x) = x^T A x$ with $$A = \begin{bmatrix} 2 & 1 \\ 1 & 2 \end{bmatrix}. $$ Then the minimum is at $x^*=(0,0)$. At $y = (1, -2)^T$, we have that $e_1^T \nabla f(y) = 2(Ay)_1 = 0$. Thus in this case $y_1 \neq x_1^*$.