Given that $f$ is continuous on $[0,1]$, differentiable on $(0,1)$, and $f(0)=0$, $f(1)=1$
Prove that there are $0<c_1 < c_2<1$ s.t. $\frac{1}{g'(c_1)}+\frac{1}{g'(c_2)}=2$ . The same for $g'(c_1)g'(c_2)=1$.