Example of a non-polynomial function $f: \mathbb{R} \to \mathbb{R}$ such that $f'(x)$ is negative for $x<0$ and positive for $x \ge 0$.

Question

I have a bunch of polynomial functions example easily (e.g. $x^2$), but have trouble coming up with a non-polynomial function.

I was thinking of defining $f(x) = e^{-x}$ for $x<0$ and $f(x) = e^{x}$ for $x \ge 0$. But it suffers the same problem as $g(x) = |x|$ because derivative does not exist at $x = 0$.

Is there any hint for this?

Answer 1

By a theorem of Darboux, a derivative has the Intermediate Value Property. So it cannot be positive at $0$ and negative somewhere else without being $0$ somewhere in between.

Answer 2

What about? $\mathbb{R} \rightarrow \mathbb{R}$, $f(x)=e^{x^2}$