To derive a formula for an approximation of $f'(x_0)$ using Taylor's theorem, in Section 4.2, pp.$188-190$ of the book Numerical Analysis by Burden and Faires, they use the following arguments:
$$3f'(x_0)=\frac{2}{h}[f(x_0 + h) - f(x_0 - h)] -\frac{1}{4h}[f(x_0 + 2h) - f(x_0 - 2h)]-\frac{h^4}{30}f^{(5)}(\xi_1)+\frac{2h^4}{15}f^{(5)}(\xi_2), $$ $\xi_1, \xi_2\in [x_0-2h,x_0+2h]$ come from the Intermediate Value Theorem.
Even if $f^{(5)}$ is continuous on $[x_0 — 2h, x_0 + 2h]$, the Intermediate Value Theorem cannot be applied because here we have the difference of terms involving $f^{(5)}$. However, an alternative method can be used to show that $f^{(5)}(\xi_1)$ and $f^{(5)}(\xi_2)$ can still be replaced by a common value $f^{(5)}(\xi)$, yielding $$f'(x_0)=\frac{1}{12h}[8f_(x_0 + h) - 8f(x_0 - h)-f(x_0 + 2h) + f(x_0 - 2h)]+\frac{h^4}{30}f^{(5)}(\xi).$$
My question: I tried to figure out what 'the alternative method' they mentioned above is? Could you please explain why they can get the existence of the number $\xi$ here? Thank you so much.