Calculus Problem by LOGA 2

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function such that $f(0)=0$ and $f(1)=1$ . Is the following statement true?

STATEMENT : For any positive integer $n$ , there exist distinct $c_1,c_2, \ldots , c_n\in(0,~1)$ such that $\frac{1}{f'(c_1)}+\frac{1}{f'(c_2)}+\cdots+\frac{1}{f'(c_n)}=n$