# Convex function with unique critical point is coercive

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?

• Your reference already contains a proof for the finite-dimensional situation... – gerw Apr 11 '18 at 6:37
• @gerw let me read It with more attention – Javier Apr 11 '18 at 6:41