By sketching a suitable pair of graphs, show that the equation
$$2 - x = \ln x$$
has only one root.
Verify by calculation that this root lies between 1.4 and 1.7.
Show that this root also satisfies the equation
$$x = \frac { 1 } { 3 } ( 4 + x - 2 \ln x )$$
Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \left( 4 + x _ { n } - 2 \ln x _ { n } \right)$$
with initial value \(x _ { 1 } = 1.5\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.