I have a specific, for-learning-sake-only question on how the author of this link:


http://www.math.ucla.edu/~yanovsky/T..._solutions.pdf


gets past the details of the Intermediate Value Theorem on the following paragraph. If someone could fill in the details for me, it would be greatly appreciated because I'm having a hard time understanding.


\begin{align}<br />
\left(\frac{3}{12h}f^{(5)}(\xi_1)+\frac{18}{12h}f^  {(5)}(\xi_2)-32\frac{6}{12h}f^{(5)}(\xi_3)+243\frac{1}{12h}f^{(  5)}(\xi_4)\right)\frac{h^5}{120}&= \\<br />
\left(\frac{3}{12}f^{(5)}(\xi_1)+\frac{18}{12}f^{(  5)}(\xi_2)-32\frac{6}{12}f^{(5)}(\xi_3)+243\frac{1}{12}f^{(5)  }(\xi_4)\right)\frac{h^4}{120}&= \\<br />
6f^{(5)}(\xi)\frac{h^4}{120}&= \\<br />
\frac{h^4}{20}f^{(5)}(\xi)<br />
\end{align}<br />

"Note that the IVT was used above..."


Shouldn't it be


"Suppose f^{(5)} is continuous on [x_0-h,x_0+3h] with
x_0-h<\xi_1<x_0<\xi_2<x_0+h<\xi_3<x_0+2h<\xi_4<x_0+3h. Since \frac{1}{4}[f^{(5)}(\xi_1)+f^{(5)}(\xi_2)+f^{(5)}(\xi_3)+f^{(5  )}(\xi_4)] is between [TEX]f^{(5)}(\xi_1)[\TEX] and f^{(5)}(\xi_4), the Intermediate Value Theorem implies that a number \xi exists between \xi_1 and \xi_4, and hence in (x_0-h,x_0+3h), with f^{(5)}(\xi)=\frac{1}{4}[f^{(5)}(\xi_1)+f^{(5)}(\xi_2)+f^{(5)}(\xi_3)+f^{(5  )}(\xi_4)]"?