7.
$$f ( x ) = 3 x ^ { 3 } - 2 x - 6$$
- Show that \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), between \(x = 1.4\) and \(x = 1.45\)
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as
$$x = \sqrt { } \left( \frac { 2 } { x } + \frac { 2 } { 3 } \right) , \quad x \neq 0$$
- Starting with \(x _ { 0 } = 1.43\), use the iteration
$$x _ { \mathrm { n } + 1 } = \sqrt { } \left( \frac { 2 } { x _ { \mathrm { n } } } + \frac { 2 } { 3 } \right)$$
to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 4 decimal places.
- By choosing a suitable interval, show that \(\alpha = 1.435\) is correct to 3 decimal places.