4.
$$\mathrm { f } ( x ) = 25 x ^ { 2 } \mathrm { e } ^ { 2 x } - 16 , \quad x \in \mathbb { R }$$
- Using calculus, find the exact coordinates of the turning points on the curve with equation \(y = \mathrm { f } ( x )\).
- Show that the equation \(\mathrm { f } ( x ) = 0\) can be written as \(x = \pm \frac { 4 } { 5 } \mathrm { e } ^ { - x }\)
The equation \(\mathrm { f } ( x ) = 0\) has a root \(\alpha\), where \(\alpha = 0.5\) to 1 decimal place.
- Starting with \(x _ { 0 } = 0.5\), use the iteration formula
$$x _ { n + 1 } = \frac { 4 } { 5 } \mathrm { e } ^ { - x _ { n } }$$
to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
- Give an accurate estimate for \(\alpha\) to 2 decimal places, and justify your answer.