Given two strictly concave, strictly increasing and everywhere derivable functions $f,g: \mathbb{R}^+_0 \to [0,1]$ where $f(0)=g(0)=0$ and $$\lim_{x\to\infty} f(x)=\lim_{x\to\infty} g(x)=1$$ Excluding $x=0$, what is the maximum number of the other intersections between them?
I think $2$ points, but I did not find a good proof of that.
Let \begin{align} f(x) &= 1 - \mathrm{e}^{-x}, \\ g(x) &= 1 - \mathrm{e}^{-x} + \frac{1}{4}\mathrm{e}^{-x}\sin x. \end{align} We have $f(0) = g(0) = 0$ and $\lim_{x\to \infty} f(x) = \lim_{x\to \infty} g(x) = 1$. Also, we have, on $[0, \infty)$,
\begin{align} f'(x) &= \mathrm{e}^{-x} > 0, \\ f''(x) &= -\mathrm{e}^{-x} < 0, \\ g'(x) &= \mathrm{e}^{-x} - \frac{1}{4}\mathrm{e}^{-x}\sin x + \frac{1}{4}\mathrm{e}^{-x}\cos x > 0\\ g''(x) &= -\mathrm{e}^{-x} - \frac{1}{2}\mathrm{e}^{-x}\cos x < 0. \end{align} Thus, $f(x)$ and $g(x)$ are both strictly concave, strictly increasing.
However, $f(x) = g(x)$ for $x = \pi, 2\pi, 3\pi, \cdots$.