Prove there exist $0\le c_1<c_2<\dots<c_n\le1$ where $\sum_{k=1}^n\frac1{f'(c_k)}=n$
For f continuous on $[0,1]$, show that there exist points $\alpha_k$ such that $\sum \limits_{k=1}^n \frac{1}{f'(\alpha_k)} = n $