(i)[3]
By using a suitable pair of graphs, show that the equation $e^{2x} = 14 - x^2$ has exactly two real roots.
(ii)[2]
Show by calculation that the positive root lies between $1.2$ and $1.3$.
(iii)[1]
Show that this root also satisfies the equation $x = \tfrac{1}{2}\ln(14 - x^2)$.
(iv)[3]
Use an iteration process based on the equation in part (iii), with a suitable starting value, to find the root correct to 2 decimal places. Record the outcome of each step to 4 decimal places.