- A curve has equation \(y = \mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = x ^ { 4 } - 5 x ^ { 2 } + 4 x - 7 \quad x \in \mathbb { R }$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval [2,3]
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as
$$x = \sqrt [ 3 ] { \frac { 5 x ^ { 2 } - 4 x + 7 } { x } }$$
The iterative formula
$$x _ { n + 1 } = \sqrt [ 3 ] { \frac { 5 x _ { n } ^ { 2 } - 4 x _ { n } + 7 } { x _ { n } } }$$
is used to find \(\alpha\)
- Starting with \(x _ { 1 } = 2\) and using the iterative formula,
- find, to 4 decimal places, the value of \(x _ { 2 }\)
- find, to 4 decimal places, the value of \(\alpha\)