5.
$$f ( x ) = 2 x ^ { 3 } - x - 4$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as
$$x = \sqrt { \left( \frac { 2 } { x } + \frac { 1 } { 2 } \right) }$$
The equation \(2 x ^ { 3 } - x - 4 = 0\) has a root between 1.35 and 1.4.
- Use the iteration formula
$$x _ { n + 1 } = \sqrt { } \left( \frac { 2 } { x _ { n } } + \frac { 1 } { 2 } \right)$$
with \(x _ { 0 } = 1.35\), to find, to 2 decimal places, the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\).
The only real root of \(\mathrm { f } ( x ) = 0\) is \(\alpha\).
- By choosing a suitable interval, prove that \(\alpha = 1.392\), to 3 decimal places.