1. ## Newton's Method

Hey guys, I'm having a problem with a concept question about Newton's Method. The question is, "Suppose your first guess is lucky, in the sense that x0 is a root of f(x) = 0. Assuming that f'( x0) is defined and not 0, what happens to x1 and later approximations?"
I think it's just the way that the question is worded that's the confusing part, but I can't seem to figure it out. Can you guys help me get started? Thanks in advance!

EDIT: x0 is x subscript 0 and x1 is x subscript 1. I dunno how to actually do it on here.

2. Once You have $\displaystyle x_{0}$ and You are able to compute $\displaystyle f(x_{0})$ and $\displaystyle f^{'}(x_{0})$, $\displaystyle x_{1}$ is given by...

$\displaystyle x_{1} = x_{0} - \frac{f(x_{0})}{f^{'}(x_{0})}$ (1)

If You are 'lucky' and $\displaystyle x_{0}$ is a 'root' of $\displaystyle f(*)$ is $\displaystyle f(x_{0})=0$ so that $\displaystyle x_{1}=x_{0}$ and the iterations are finished!...

In general the choice of $\displaystyle x_{0}$ is the most critical step in Newton's method and if You are 'not lucky' may be that the algorithm doesn't converge ...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$