By sketching a suitable pair of graphs, show that the equation
$$3 \mathrm { e } ^ { x } = 8 - 2 x$$
has only one root.
Verify by calculation that this root lies between \(x = 0.7\) and \(x = 0.8\).
Show that this root also satisfies the equation
$$x = \ln \left( \frac { 8 - 2 x } { 3 } \right)$$
Use the iterative formula \(x _ { n + 1 } = \ln \left( \frac { 8 - 2 x _ { n } } { 3 } \right)\) to determine this root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.