Example of a concave function that grows faster than a polynomial function as $x$ goes to $-\infty$ and have larger values as well

Question

Let us think of a polynomial function $-x^3$.

Is there a concave function that is positive and grows faster than this polynomial function as $x$ goes to $-\infty$?

I cannot find a good explicit example myself, though I can vaguely imagine one...Could anyone help me?

Answer 1

There cannot be such a function. If $f$ is positive and concave then for $x < -1$ $$ 0 < f(x) \le L(x) $$ where $L$ is the secant line joining $(-1, f(-1))$ and $(0, f(0))$, which means that $f$ grows at most linear for $x \to -\infty$.