1. ## Rolle's Theorm example

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 $x\in[x_{i-1},x_{i}],$ subinterval of data set $D=\{(x_{i},y_{i}=f(x_{i}))\}, i=0,1,...N$ there exists some $c\in [x_{i-1},x_{i}]$ such that
$\\f'(x)-S'(x) = \int_{c}^{x} [f''(t)-S''(t)]dt$

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.

2. ## Re: Rolle's Theorm example

Originally Posted by cyndiblock
Show, using Rolle's Theorm, that for every $x\in[x_{i-1},x_{i}],$ subinterval of data set $D=\{(x_{i},y_{i}=f(x_{i}))\}, i=0,1,...N$ there exists some $c\in [x_{i-1},x_{i}]$ such that
$\\f'(x)-S'(x) = \int_{c}^{x} [f''(t)-S''(t)]dt$
What is $S$ ?. What are the hypothesis for $f$ ?.

3. ## Re: Rolle's Theorm example

so S is the cubic spline interpolation of D, then we know that $S_i=y_i, i=1,2,....,N$ and $S_i'(x_i)=S_{i+1}'(x_i), i=1,...N-1$ and $S_i''(x_i)=S_{i+1}''(x_i), i =1, ...., N-1$ .... does that help me? I'm just really confused with this stuff, thanks for your time.