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When deriving the five-point differentiation formula as shown in this book, the IVT was used to combine $ f^{(5)} (\xi_1) $ and $ f^{(5)} (\xi_2) $ into one error term, $ f^{(5)}(\tilde{\xi}) $

As the book says here, another method is used to combine $ f^{(5)}(\tilde{\xi}) $ and $ f^{(5)}(\hat{\xi}) $.

What is the method used here?

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What's the relation between $\xi_1$, $\xi_2$, $\tilde{\xi}$ and $\hat{\xi}$? – Robert Israel Dec 10 '12 at 1:32

You're still using IVT, just in a way that's a little more subtle.

You want to show that there exists $\xi$ such that $4f^{(5)}(\hat{\xi}) - f^{(5)}(\tilde{\xi}) = 3f^{(5)}(\xi)$.

Or, written differently, show exists $c$ such that $f(a) + \frac{f(a) - f(b)}{3} = f(c)$. And this is straightforward by IVT.

Edit: No, sorry I was wrong. You can't use IVT to solve this. And in fact, this isn't true in general. Burden/Faires is assuming that $h$ is sufficiently small to make it work by continuity of $f$. But it seems to me that this proof won't work when $f(a)$ is a positive local maximum of $f$ without making $h = 0$.

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