2.
$$f ( x ) = x ^ { 3 } - 5 x + 16$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as
$$x = ( a x + b ) ^ { \frac { 1 } { 3 } }$$
giving the values of the constants \(a\) and \(b\).
The equation \(\mathrm { f } ( x ) = 0\) has exactly one real root \(\alpha\), where \(\alpha = - 3\) to one significant figure.
- Starting with \(x _ { 1 } = - 3\), use the iteration
$$x _ { n + 1 } = \left( a x _ { n } + b \right) ^ { \frac { 1 } { 3 } }$$
with the values of \(a\) and \(b\) found in part (a), to calculate the values of \(x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving all your answers to 3 decimal places.
- Using a suitable interval, show that \(\alpha = - 3.17\) correct to 2 decimal places.