1 The curve \(y = 3 ^ { x }\) intersects the curve \(y = 10 - x ^ { 3 }\) at the point where \(x = \alpha\).
- Show that \(\alpha\) lies between 1 and 2 .
- Show that the equation \(3 ^ { x } = 10 - x ^ { 3 }\) can be rearranged into the form
$$x = \sqrt [ 3 ] { 10 - 3 ^ { x } }$$
- Use the iteration \(x _ { n + 1 } = \sqrt [ 3 ] { 10 - 3 ^ { x _ { n } } }\) with \(x _ { 1 } = 1\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.