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