MHB Intermediate Value Thm for Five-Point Formula

AI Thread Summary
The discussion centers on a question regarding the application of the Intermediate Value Theorem (IVT) in a mathematical context related to a five-point formula. The original poster seeks clarification on how the author of a linked document employs the IVT in their solution. A reminder is issued about the importance of notifying forum members when questions are posted on multiple platforms to avoid redundant efforts. The original poster later concludes that the problem is unnecessary for their goals and wishes to close the discussion. This highlights the importance of clear communication and understanding in mathematical problem-solving.
kalish1
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I have a specific, for-learning-sake-only question on how the author of this link:

http://www.math.ucla.edu/~yanovsky/Teaching/Math151A/hw5/Hw5_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.

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

"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 $f^{(5)}(\xi_1)$ 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 $f^{(5)}(\xi)=\frac{1}{4}[f^{(5)}(\xi_1)+f^{(5)}(\xi_2)+f^{(5)}(\xi_3)+f^{(5)}(\xi_4)]$"?
 
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Hi kalish,

Here's a friendly reminder about this part of Rule #2 here on MHB.

Snippet from MHB Rule #2 said:
As a courtesy, if you post your problem on multiple websites, and you get a satisfactory response on a different website, indicate in your MHB thread that you got an answer elsewhere so that our helpers do not duplicate others' efforts.

As a courtesy (not just to us, but also to those on other sites as well), you should inform our (and their) members when you've posted a question on multiple sites (for instance, I see that you've asked this same question on math.stackexchange) and then inform us if you find a solution elsewhere. This way, no one's efforts are duplicated and/or put to waste on solving a problem that has been potentially solved. (Smile)
 
Chris L T521 said:
Hi kalish,

Here's a friendly reminder about this part of Rule #2 here on MHB.
As a courtesy (not just to us, but also to those on other sites as well), you should inform our (and their) members when you've posted a question on multiple sites (for instance, I see that you've asked this same question on math.stackexchange) and then inform us if you find a solution elsewhere. This way, no one's efforts are duplicated and/or put to waste on solving a problem that has been potentially solved. (Smile)

Thanks Chris. I decided that this problem was unnecessary because the goal I am trying to achieve doesn't actually make sense. So I would like to close this post.
 
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