By sketching a suitable pair of graphs, show that the equation \(x ^ { 3 } = 3 - x\) has exactly one real root.
Show that if a sequence of real values given by the iterative formula
$$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 3 } + 3 } { 3 x _ { n } ^ { 2 } + 1 }$$
converges, then it converges to the root of the equation in part (i).
Use this iterative formula to determine the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.