11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fa4afaf4-fe5d-4f3a-b3de-9600d5502a49-32_589_771_248_648}
\captionsetup{labelformat=empty}
\caption{Figure 8}
\end{figure}
Figure 8 shows a sketch of the curve \(C\) with equation \(y = x ^ { x } , x > 0\)
- Find, by firstly taking logarithms, the \(x\) coordinate of the turning point of \(C\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)
The point \(P ( \alpha , 2 )\) lies on \(C\). - Show that \(1.5 < \alpha < 1.6\)
A possible iteration formula that could be used in an attempt to find \(\alpha\) is
$$x _ { n + 1 } = 2 x _ { n } ^ { 1 - x _ { n } }$$
Using this formula with \(x _ { 1 } = 1.5\)
- find \(x _ { 4 }\) to 3 decimal places,
- describe the long-term behaviour of \(x _ { n }\)