Convex function with unique critical point is coercive

Question

Let $V:\mathbb{R}^d \to \mathbb{R} \in \mathcal{C}^1$ such that $V$ is convex and $V$ has a unique critical point.

Then $V$ is coercive.

This was an example given in one my lectures. But it sounds like one of this examples for which there might a counterexample due to non-formal statement...

Can you prove that is right? Or there is such a counter-example?

References

The similar questions window tells me that in fact For a convex function, does having a unique minimizer imply that it is coercive? is not true in general Hilbert space. What about finite-dimensional ones?

Answer 1

Yes, that statement is correct but the proof is not so easy. See Corollary 8.7.1 in Rockafellar's Convex Analysis.