Existence of unique maximizer in $\mathbb{R}^n$

Question

This sounds like a very basic question, but I have a hard time pinpointing the necessary and sufficient conditions...

Let $f:\mathbb{R}^n \to \mathbb{R}$ be a function. I want to prove that there is a unique maximizer $\pmb{x}^\star \in \mathbb{R}^n$. What I know:

• $f$ is differentiable
• $f$ is strictly (but not strongly) concave
• $f$ is bounded from above

As I see it, this is enough to say that there is a unique maximizer, but it is not enough to say that this maximizer is in $\mathbb{R}^n$. What condition should I add such that I can guarantee that this is the case?

As an example, for $n = 1$, consider the function $$f(x) = ax - e^x, \quad a \ge 0.$$

the function is strictly concave and bounded by above for any $a$, however:

• for $a > 0$, it has a maximizer $x^\star \in \mathbb{R}$
• for $a = 0$, the "maximizer" is $-\infty \not\in \mathbb{R}$.

So I suppose a summary of the question is: what condition, in addition to differentiability, strict concavity and upper-boundedness, should a function have in order to ensure existence and uniqueness of a maximizer?

Answer 1

A stationary point in $\mathbb{R}^n$ is sufficient.