For twice differentiable convex function $f(x)$ on $\mathbb{R}^n$, with $x=\langle x_1,x_2,...,x_n \rangle$,
prove that, $\forall x \in \mathbb{R}^n$ and $\forall i \in \{1,2,...,n\}$, if $\frac{\partial f(x) }{\partial{} ~x_i} < 0$, then
$$x^*_i \ge x_i$$
where $x^*=\langle x^*_1,x^*_2,...,x^*_n \rangle$ is the extreme point (global minimum) of convex function $f$.
A counterexample for two dimension convex function, picture shows the contour lines, where all rings are perfect circle!
A concrete example would be $f(x,y)=x^{2}+\left(y+x\right)^{2}$