I'm looking for example of functions defined on $[0,+\infty)$ with the following properties:
1) continuous, twice differentiable
2) $f(0)=0$, $\lim_{x\to+\infty} f(x)=1/3$
3) $f^{\prime}>0$, $f^{\prime \prime}<0$
4) $f^{\prime}(0)=+\infty$, $\lim_{x\to +\infty} f^{\prime}(x)=0$
Do you have some examples?
The functions $f_1(x) = \dfrac{1}{3}\dfrac{e^{\sqrt{x}}-1}{e^{\sqrt{x}}+1}$, and $f_2(x) = \dfrac{2}{3\pi}\arctan(\sqrt{x})$ are both good examples.
Of course, there are infinitely many others out there.