Intersection between two increasing concave functions

Question

Let $f,g:[0,1]\mapsto[0,1]$ becontinuously increasing concave functions such that $f(0)=g(0)=0$ and $f(1)=g(1)=1$. I wish to know what is the maximum intersection points between $f$ ad $g$. I have a conjecture that the only intersection points are only $(0,0)$ and $(1,1)$. Is it true or there is any counterexample?

Answer 1

It is not true. Take the two piecewise linear continuous functions

$f$ passing by $(1/9,4/9), (5/9,8/9)$

$g$ passing by $(2/9,6/9)$.

They cross at $(2/9,5/9)$ and $(4/9,7/9)$.

Using regular polygons centered in $(1,0)$ you may find as many points of intersection as you desire.