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Math Help - Rolle's Theorm example

  1. #1
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    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.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Rolle's Theorm example

    Quote Originally Posted by cyndiblock View Post
    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 ?.
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  3. #3
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    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.
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