2.
$$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 3 x - 11$$
- Show that \(\mathrm { f } ( x ) = 0\) can be rearranged as
$$x = \sqrt { } \left( \frac { 3 x + 11 } { x + 2 } \right) , \quad x \neq - 2 .$$
The equation \(\mathrm { f } ( x ) = 0\) has one positive root \(\alpha\).
The iterative formula \(x _ { n + 1 } = \sqrt { } \left( \frac { 3 x _ { n } + 11 } { x _ { n } + 2 } \right)\) is used to find an approximation to \(\alpha\).
- Taking \(x _ { 1 } = 0\), find, to 3 decimal places, the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\).
- Show that \(\alpha = 2.057\) correct to 3 decimal places.