I am looking for a function $f(x)$ where
- $f(x)\rightarrow\infty$ for $x\rightarrow0$
- $f(x)=0$ for $x=1$
- $f(x)\rightarrow\infty$ for $x\rightarrow\infty$
- only extrempoint (minimum obviously) is at $x=1$
- $f$ is completely differentiable everywhere for $x>0$
I am sure there is a standard way to express a function like that with the Euler constant.
I want to use the function as a cost function for artificial life objects to keep a distance of 1. It should be infinitely costly for two objects to be at the same position and also to move far away from each other.