Let $f\in C^1$ convex defined in a convex $S$. If there exists $x^*\in S$ such that for all $y\in S$ it is true that $$\nabla^t f(x^*)(y-x^*)\ge 0$$ then $x^*$ is a global minimizer of $f$ in $S$
I know from here $f$ convex $\iff$ $f(y)\ge f(x)+\nabla f(x)(x-y)$ so we know that for this particular $x^*$ and any $y$ we have
$$f(y)\ge f(x^*) + \nabla f(x^*)(y-x^*) \implies f(y)-f(x^*)\ge \nabla f(x^*)(y-x^*)\ge 0\implies f(y)\ge f(x^*)$$
for any $y$, which means $x^*$ is a global minimizer.
Can you spot any errors in my proof?