Example of a function with $f^\prime (x)=f^\prime(1-x)$

Question

Let $f:[0,1]\rightarrow [0,\infty)$ be a strictly descending continuous function such that $f^\prime(x)=f^\prime(1-x)$ for all $x$. Obviously $f(x)=1-x$ is an example of these functions. Is there another example? Can we characterize all of these functions? Thanks a lot.

Answer 1

$f'(x+1/2)$ is an even function on $[-1/2, 1/2]$, so $f(x+1/2)-f(1/2)$ is an odd function. Any strictly decreasing, differentiable odd function $g$ on $[-1/2, 1/2]$ will do, and then take $f(t) = g(t - 1/2) + c$ where $c \ge -g(1/2)$.