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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

  • 0
    This is too old to migrate, but you might want to delete and repost on [Computational Science](http://scicomp.stackexchange.com/)2013-01-23

0 Answers 0