3 The curve \(y = \cos ^ { - 1 } ( 2 x - 1 )\) intersects the curve \(y = \mathrm { e } ^ { x }\) at a single point where \(x = \alpha\).
- Show that \(\alpha\) lies between 0.4 and 0.5 .
- Show that the equation \(\cos ^ { - 1 } ( 2 x - 1 ) = \mathrm { e } ^ { x }\) can be written as \(x = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x } \right)\).
- Use the iteration \(x _ { n + 1 } = \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos \left( \mathrm { e } ^ { x _ { n } } \right)\) with \(x _ { 1 } = 0.4\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.