5 The curve with equation
$$y = 5 \mathrm { e } ^ { 2 x } - 8 x ^ { 2 } - 20$$
crosses the \(x\)-axis at only one point. This point has coordinates \(( p , 0 )\).
- Show that \(p\) satisfies the equation \(x = \frac { 1 } { 2 } \ln \left( 1.6 x ^ { 2 } + 4 \right)\).
- Show by calculation that \(0.75 < p < 0.85\).
- Use an iterative formula based on the equation in part (i) to find the value of \(p\) correct to 5 significant figures. Give the result of each iteration to 7 significant figures.
- Find the gradient of the curve at the point \(( p , 0 )\).