It is given that \(a\) and \(b\) are positive constants. By sketching graphs of
$$y = x ^ { 5 } \quad \text { and } \quad y = a - b x$$
on the same diagram, show that the equation
$$x ^ { 5 } + b x - a = 0$$
has exactly one real root.
Use the iterative formula \(x _ { n + 1 } = \sqrt [ 5 ] { 53 - 2 x _ { n } }\), with a suitable starting value, to find the real root of the equation \(x ^ { 5 } + 2 x - 53 = 0\). Show the result of each iteration, and give the root correct to 3 decimal places.