Question: Let $f$ be twice differentiable on $[a,b]$, $f(a) < 0,$ $f(b) > 0,$ $f'(x) > \delta,$ and $0 \leq f''(x) \leq M \;\;\forall x \in [a,b]$.
Let $\xi$ be the unique point in $(a,b)$ at which $f(\xi) = 0$.
(i) Show that Newton’s method amounts to finding a fixed point of the function $g$ defined by
$g(x)=x-\frac{f(x)}{f'(x)}$.
How does $g'(x)$ behave for $x$ near $\xi$ ?
# I don't know how to proceed for this question. I need help in solving above 2 questions. Thank you.