Is it possible for strictly increasing and strictly concave-up functions to cut each other twice or more?

Question

Assume that two positive functions are strictly increasing and strictly concave-up, i.e. first and second derivatives are positive. Both functions start from the point $(0,0)$. Then, is it possible for them to cut each other twice or more? If so, can someone please bring a counterexample that shows this?

Answer 1

Consider $f, g: (0, \infty)\rightarrow\mathbb{R}$ defined by $f(x) = x^{2}+\sin(x)$ and $g(x) = x^{2}$. They have infinitely many points of intersection.

Answer 2

Twice;

$f(x) = e^x$

$g(x) = e^{2x} + 0.1$

I don't think $> 2$ is possible but I'm unable to prove it.