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I have a function $$ F(x)= \frac{x^3 - 14x^2 + 7x + 203}{(x-3)(8-x)} $$

I need to use Newton's Method to find the max interval such that a number of constraints are valid.

$3 < a < b < 8$,

$f\in C^2[ a, b ]$,

$f\left( \frac{2}{3}a + \frac{1}{3}b\right)f\left(\frac{1}{3}a+\frac{2}{3}b\right) < 0$,

$f'(x)\neq 0$ for all $x \in [ a, b ]$,

$|f(x) f''(x)| < [f'(x)]^2$ for all $x \in (a, b)$.

I have used Newtons Method to discover that the approx root on this function is: $5.22520933956314404$

With some research, i have noted that if $e = \frac{1}{3}(b-a)$, $f$ has a root in $[a+e, b-e]$;

Using this i have verified all of the conditions in my question hold. And they do.

i have used: $a=4.66$ $b=6.33$

Question: How can i know, and prove that $[a+e, b-e]$ is the largest possible interval between $(3, 8)$?

I can provide my script if anyone is interested.

note*: Sorry for formatting, I'm still trying to figure it all out

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    What does `f C2[ a, b ]` mean? Do you mean $f \in [a,b]$?2012-04-23
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    Also does `x 2 ` and `f'(x)2` mean power 2? i.e. $x^2$ and $f'(x)^2$?2012-04-23
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    I do not think this equation was properly translated to LaTeX by @JoeJohnson: `|f(x) f''(x)| < f'(x)2 for all x in (a, b).` The `2` might be an exponent!2012-04-23
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    @J.D.: I will put it in as a power of $2$. The OP can give us guidance, maybe?2012-04-23
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    Where can i read about the formatting stuff? Everything above looks correct, thanks for formatting help2012-04-23
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    @Special--k: If you submit or edit a question, you will see a circular question-mark icon above your text, on the right. Click on that for help.2012-04-23
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    This is too old to migrate, but you might want to delete and repost on [Computational Science](http://scicomp.stackexchange.com/)2013-01-23

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