4. \(\mathrm { f } ( x ) = x ^ { 2 } + 5 x - 2\) sec \(x , \quad x \in \mathbb { R } , \quad - \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\).
- Show that the equation \(\mathrm { f } ( x ) = 0\) has a root, \(\alpha\), such that \(1 < \alpha < 1.5\)
- Show that a suitable rearrangement of the equation \(\mathrm { f } ( x ) = 0\) leads to the iterative formula
$$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 2 } { x _ { n } ^ { 2 } + 5 x _ { n } } \right)$$
- Use the iterative formula in part (ii) with a starting value of 1.25 to find \(\alpha\) correct to 3 decimal places. You should show the result of each iteration.