This is a challenge question:

Let $\displaystyle f:[0,1]\to [0,1]$ with $\displaystyle f(0)=0, f(1)=1$ differentiable

Show there exists distinct points $\displaystyle x,y\in [0,1]$ such that $\displaystyle f'(x)f'(y)=1$.

Moderator edit: Approved Challenge question.