(a)[2]
You are told that $e^{2x} = 5 + \cos 3x$ has just one root. By calculation, show that this root is within $0.7 \le x \le 0.8$.
(b)[1]
Show that any sequence of values in the interval $0.7 \le x \le 0.8$ defined by the iterative formula $x_{n+1} = \frac{1}{2} \ln(5 + \cos 3x_n)$, if it converges, converges to the root of the equation in part (a).
(c)[3]
Use this iterative procedure to find the root correct to 3 decimal places. Include each iteration result to 5 decimal places.