I want to find out a continuous function on $[0,L]$, $L$ is a positive number, which looks like this red curve:
The function is always positive.
This function firstly is strictly concave, then the function is strictly convex. Moreover, like the graph shows, this function increases rapidly first and then increases very slowly and then increases very quickly.
But I can not come up one such function. Can you help to find out a very simple continuous function to satisfy this graph ? Thank you!