This is a simple question but i am a bit confused now:
- $f(x) \sim x$ when $x \sim 0$
- $f(x) \sim 1$ when $x \to \infty$
- $f(1) \approx \frac 14$
- $|\frac{d^nf(x)}{dx^n}|$ strictly decreasing
Normally i do $f(x)=1-e^{-x}$ or $f(x)=\arctan(\frac \pi 2 x)\frac 2 \pi$, but these are too high at $f(1)$. Do you have a simple choice for adapting to the lower value at $f(1)$, having their derivatives also decreasing, just like the indicated functions?