2. \(\quad \mathrm { f } ( x ) = x ^ { 3 } - 2 x - 5\).
- Show that there is a root \(\alpha\) of \(\mathrm { f } ( x ) = 0\) for \(x\) in the interval \([ 2,3 ]\).
The root \(\alpha\) is to be estimated using the iterative formula
$$x _ { n + 1 } = \sqrt { \left( 2 + \frac { 5 } { x _ { n } } \right) } , \quad x _ { 0 } = 2$$
- Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 significant figures.
- Prove that, to 5 significant figures, \(\alpha\) is 2.0946.