1.
$$g ( x ) = x ^ { 6 } + 2 x - 1000$$
- Show that \(\mathrm { g } ( x ) = 0\) has a root \(\alpha\) in the interval [3,4]
Using the iteration formula
$$x _ { n + 1 } = \sqrt [ 6 ] { 1000 - 2 x _ { n } } \quad \text { with } x _ { 1 } = 3$$
- find, to 4 decimal places, the value of \(x _ { 2 }\)
- find, by repeated iteration, the value of \(\alpha\).
Give your answer to 4 decimal places.