4 The equation \(x ^ { 3 } - 2 x - 2 = 0\) has one real root.
- Show by calculation that this root lies between \(x = 1\) and \(x = 2\).
- Prove that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 2 } { 3 x _ { n } ^ { 2 } - 2 }$$
converges, then it converges to this root.
- Use this iterative formula to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.