Real analysis problem: Rolle's theorem, Darboux's theorem, induction I am taking a course in real analysis (undergraduate) and I'm supposed to solve a problem and present in class. I've just started, but I'm already stuck. This is the problem:  
Let a function $f$ be continuous on $[0,1]$ and differentiable on $(0,1)$. Assume that $f(0)=f(1)$. Prove that for any $n\in N$ there exist points $0<c_1<...<c_n<1$ such that $\sum_{k=1}^nf'(c_k)=0$. Hint: apply Rolle's theorem, induction and Darboux's theorem.  
This is what I have so far:
Let $g(x):=f(x)-f(1)$, then $g$ is continuous on $[0,1]$ and differentiable on $(0,1)$. $g(1)=g(0)=0$ so Rolle's theorem gives that there exists a point $c_1\in (0,1)$ s.t. $g'(c_1) =f'(c_1) =0$.  
Darboux's theorem requires that a function is differentiable on a closed interval so I need some point $c$ s.t. $0<c_1<c<1$ so that the function is differentiable on $[c_1,c]$.
I haven't gotten any further than this and would like advice on how to proceed from here.  
Update: I aksed my lecturer and he said that the first part with Rolle's theorem would be the first step in the induction proof. So then I would assume that there exist $n$ points $0<c_1<...<c_n<1$ s.t. $\sum_{k=1}^n f'(c_k)=0$ and then use that to prove that for $n+1$ points $0<c_1<...<c_n<c_{n+1}<1$ we also have that $\sum_{k=1}^{n+1}f'(c_k)=0$.
$\sum_{k=1}^{n+1}f'(c_k)=\sum_{k=1}^nf'(c_k)+f'(c_{n+1}) = f'(c_{n+1})$. If $f'(c_{n+1})=0$ then the problem is solved. But what if $f'(c_{n+1})>0$ or $f'(c_{n+1})<0$?
I guess this is where I'm supposed to use Darboux's theorem, but I'm not sure how.

I'd observe that $0=f(1)-f(0) = \sum_{k=1}^n\left[f(k/n)-f((k-1)/n)\right]$ and then apply the MVT on each interval $[(k-1)/n,k/n]$.
