4.
$$\mathrm { f } ( x ) = x ^ { 3 } + x ^ { 2 } - 4 x - 1$$
The equation \(\mathrm { f } ( x ) = 0\) has only one positive root, \(\alpha\).
- Show that \(\mathrm { f } ( x ) = 0\) can be rearranged as
$$x = \sqrt { \left( \frac { 4 x + 1 } { x + 1 } \right) } , x \neq - 1$$
The iterative formula \(x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }\) is used to find an approximation to \(\alpha\).
- Taking \(x _ { 1 } = 1\), find, to 2 decimal places, the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
- By choosing values of \(x\) in a suitable interval, prove that \(\alpha = 1.70\), correct to 2 decimal places.
- Write down a value of \(x _ { 1 }\) for which the iteration formula \(x _ { n + 1 } = \sqrt { \left( \frac { 4 x _ { n } + 1 } { x _ { n } + 1 } \right) }\) does not produce a valid value for \(x _ { 2 }\).
Justify your answer.