5.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3ff6824f-9fbf-4b5b-8bab-91332c549b36-08_624_1054_274_447}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where
$$\mathrm { f } ( x ) = ( 8 - x ) \ln x , \quad x > 0$$
The curve cuts the \(x\)-axis at the points \(A\) and \(B\) and has a maximum turning point at \(Q\), as shown in Figure 1.
- Write down the coordinates of \(A\) and the coordinates of \(B\).
- Find f'(x).
- Show that the \(x\)-coordinate of \(Q\) lies between 3.5 and 3.6
- Show that the \(x\)-coordinate of \(Q\) is the solution of
$$x = \frac { 8 } { 1 + \ln x }$$
To find an approximation for the \(x\)-coordinate of \(Q\), the iteration formula
$$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } }$$
is used.
- Taking \(x _ { 0 } = 3.55\), find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\).
Give your answers to 3 decimal places.