5.
$$f ( x ) = - x ^ { 3 } + 4 x ^ { 2 } - 6$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root between \(x = 1\) and \(x = 2\)
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be rewritten as
$$x = \sqrt { \left( \frac { 6 } { 4 - x } \right) }$$
- Starting with \(x _ { 1 } = 1.5\) use the iteration \(x _ { n + 1 } = \sqrt { \left( \frac { 6 } { 4 - x _ { n } } \right) }\) to calculate the values of \(x _ { 2 }\), \(x _ { 3 }\) and \(x _ { 4 }\) giving all your answers to 4 decimal places.
- Using a suitable interval, show that 1.572 is a root of \(\mathrm { f } ( x ) = 0\) correct to 3 decimal places.