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?