- A curve has equation \(y = \mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = x ^ { 2 } - 5 x + \mathrm { e } ^ { x } \quad x \in \mathbb { R }$$
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), in the interval [1,2]
The iterative formula
$$x _ { n + 1 } = \sqrt { 5 x _ { n } - \mathrm { e } ^ { x _ { n } } }$$
with \(x _ { 1 } = 1\) is used to find an approximate value for the root \(\alpha\).
- Find the value of \(x _ { 2 }\) to 4 decimal places.
- Find, by repeated iteration, the value of \(\alpha\), giving your answer to 4 decimal places.