prove the identity for a differentiable function

Question

Let $f$ be a smooth real function and $f(0)=0, f (1)=1$, prove that exists various $x_1$, $x_2$ $ \in [0;1] $ such that $$\frac{1}{f'(x_1)}+\frac{1}{f'(x_2)}=2$$ I tried to use the mean value theorem for different intervals, but it didn't help me. Also I can't understand the behavior of this function. Thank you for help!