Analysis I | Part IA, 2001

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

g(t)g(α)g(α)(tα)Ltα2\left|g(t)-g(\alpha)-g^{\prime}(\alpha)(t-\alpha)\right| \leq L|t-\alpha|^{2}

for all tα<δ(ϵ)|t-\alpha|<\delta(\epsilon).

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

If F(t)=t(f(t)/f(t))F(t)=t-\left(f(t) / f^{\prime}(t)\right) and a sequence {xn}\left\{x_{n}\right\} is defined by b=x0,xn=b=x_{0}, x_{n}= F(xn1)(n>0)F\left(x_{n-1}\right) \quad(n>0), show that x0,x1,x2,x_{0}, x_{1}, x_{2}, \ldots is a decreasing sequence of points in [a,b][a, b] and hence has limit α\alpha. What is f(α)f(\alpha) ? Using part (i) or otherwise estimate the rate of convergence of xnx_{n} to α\alpha, i.e., the behaviour of the absolute value of (xnα)\left(x_{n}-\alpha\right) for large values of nn.

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