1.II.11D

(i) Show that if $g: \mathbb{R} \rightarrow \mathbb{R}$ is twice continuously differentiable then, given $\epsilon>0$, we can find some constant $L$ and $\delta(\epsilon)>0$ such that

$\left|g(t)-g(\alpha)-g^{\prime}(\alpha)(t-\alpha)\right| \leq L|t-\alpha|^{2}$

for all $|t-\alpha|<\delta(\epsilon)$.

(ii) Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be twice continuously differentiable on $[a, b]$ (with one-sided derivatives at the end points), let $f^{\prime}$ and $f^{\prime \prime}$ be strictly positive functions and let $f(a)<0.

If $F(t)=t-\left(f(t) / f^{\prime}(t)\right)$ and a sequence $\left\{x_{n}\right\}$ is defined by $b=x_{0}, x_{n}=$ $F\left(x_{n-1}\right) \quad(n>0)$, show that $x_{0}, x_{1}, x_{2}, \ldots$ is a decreasing sequence of points in $[a, b]$ and hence has limit $\alpha$. What is $f(\alpha)$ ? Using part (i) or otherwise estimate the rate of convergence of $x_{n}$ to $\alpha$, i.e., the behaviour of the absolute value of $\left(x_{n}-\alpha\right)$ for large values of $n$.