1.
$$f ( x ) = x ^ { 5 } + x ^ { 3 } - 12 x ^ { 2 } - 8 , \quad x \in \mathbb { R }$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as
$$x = \sqrt [ 3 ] { \frac { 4 \left( 3 x ^ { 2 } + 2 \right) } { x ^ { 2 } + 1 } }$$
- Use the iterative formula
$$x _ { n + 1 } = \sqrt [ 3 ] { \frac { 4 \left( 3 x _ { n } ^ { 2 } + 2 \right) } { x _ { n } ^ { 2 } + 1 } }$$
with \(x _ { 0 } = 2\), to find \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) giving your answers to 3 decimal places.
The equation \(\mathrm { f } ( x ) = 0\) has a single root, \(\alpha\).
- By choosing a suitable interval, prove that \(\alpha = 2.247\) to 3 decimal places.