It is given that \(k\) is a positive constant. By sketching the graphs of
$$y = 14 - x ^ { 2 } \text { and } y = k \ln x$$
on a single diagram, show that the equation
$$14 - x ^ { 2 } = k \ln x$$
has exactly one real root.
The real root of the equation \(14 - x ^ { 2 } = 3 \ln x\) is denoted by \(\alpha\).
(a) Find by calculation the pair of consecutive integers between which \(\alpha\) lies.
(b) Use the iterative formula \(x _ { n + 1 } = \sqrt { 14 - 3 \ln x _ { n } }\), with a suitable starting value, to find \(\alpha\). Show the result of each iteration, and give \(\alpha\) correct to 2 decimal places.