$f(0) = 0, f(1) = 1$. Prove that $\frac{1}{f'(x_1)} + \frac{1}{f'(x_2)} = 2$ for some $x_1, x_2 \in [0, 1]$
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 $