Re: Rolle's Theorm example

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**cyndiblock** 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

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

What is $\displaystyle S$ ?. What are the hypothesis for $\displaystyle f$ ?.

Re: Rolle's Theorm example

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