Past exam of the mathematics course of the University of Cambridge / 2024 / ib / Paper 4 / 18H / b

Table of contents
- Solution b

Solution

If

x_{k} \geq 1

, then positivity and the arithmetic-geometric mean inequality give

x_{k + 1} = \frac{1}{2} (x_{k} + \frac{a}{x _{k}}) \geq a \geq 1.

Thus all iterates remain in

[1, \infty)

.

Consider the strictly convex function

f (x) = \frac{x ^{3}}{3} - a x

on

[1, \infty)

. Since

f^{'} (x) = x^{2} - a

and

f^{''} (x) = 2 x

, its Newton minimization step is

x - \frac{f ^{'} ( x )}{f ^{''} ( x )} = \frac{1}{2} (x + \frac{a}{x}) .

Its unique minimizer is

x^{*} = a

.

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