(a)[2]
By sketching an appropriate pair of graphs, show that the equation $\ln x = 3x - x^2$ has one real root.
(b)[2]
Use calculation to verify that the root is between $2$ and $2.8$.
(c)[3]
Apply the iterative formula $x_{n+1} = \sqrt{3x_n - \ln x_n}$ to find the root correct to $2$ decimal places. Record the outcome of every iteration to $4$ decimal places.