6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{659a0479-c8c6-418b-b8a9-67ad68474023-12_624_1057_258_504}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch 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 2.
- Find the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\).
- Show that the \(x\) coordinate of \(Q\) satisfies
$$x = \frac { 8 } { 1 + \ln x }$$
- Show that the \(x\) coordinate of \(Q\) lies between 3.5 and 3.6
- Use the iterative formula
$$x _ { n + 1 } = \frac { 8 } { 1 + \ln x _ { n } } \quad n \in \mathbb { N }$$
with \(x _ { 1 } = 3.5\) to
- find the value of \(x _ { 5 }\) to 4 decimal places,
- find the \(x\) coordinate of \(Q\) accurate to 2 decimal places.