It is given that the curves with equations \(y = 6 \ln x\) and \(y = 8 x - x ^ { 2 } - 3\) intersect at a single point where \(x = \alpha\).
Show that \(\alpha\) lies between 5 and 6 .
Show that the equation \(x = 4 + \sqrt { 13 - 6 \ln x }\) can be rearranged into the form
$$6 \ln x + x ^ { 2 } - 8 x + 3 = 0$$
Use the iterative formula
$$x _ { n + 1 } = 4 + \sqrt { 13 - 6 \ln x _ { n } }$$
with \(x _ { 1 } = 5\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\), giving your answers to three decimal places.
A curve has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x ) = 6 \ln x + x ^ { 2 } - 8 x + 3\).
Find the exact values of the coordinates of the stationary points of the curve.
Hence, or otherwise, find the exact values of the coordinates of the stationary points of the curve with equation
$$y = 2 \mathrm { f } ( x - 4 )$$