By sketching a suitable pair of graphs, show that the equation \(\ln x = 3 x - x ^ { 2 }\) has one real root.
Verify by calculation that the root lies between 2 and 2.8.
Use the iterative formula \(x _ { n + 1 } = \sqrt { 3 x _ { n } - \ln x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.