I am a student at SMU and I have been working on this problem for a couple of hours and can't seem to figure it out. This question has a few parts to it, but the later parts rely on the first one, so maybe if I can get some help on that I will be able to do the others. Here is the question:

Show, using Rolle's Theorm, that for every $\displaystyle x\in[x_{i-1},x_{i}], $ subinterval of data set $\displaystyle D=\{(x_{i},y_{i}=f(x_{i}))\}, i=0,1,...N$ there exists some $\displaystyle c\in [x_{i-1},x_{i}]$ such that

$\displaystyle \\f'(x)-S'(x) = \int_{c}^{x} [f''(t)-S''(t)]dt$

I can see why this would be the case from the MVT, but I'm having a difficult time formulating a proof. Help would be greatly appreciated, then *hopefully* I will be able to get somewhere with the next part to this problem.