2 For $0 < x \leqslant 2$, the curves with equations $y = 4 \ln x$ and $y = \sqrt { x }$ intersect at a single point where $x = \alpha$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\alpha$ lies between 0.5 and 1.5.
\item Show that the equation $4 \ln x = \sqrt { x }$ can be rearranged into the form

$$x = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }$$
\item Use the iterative formula

$$x _ { n + 1 } = \mathrm { e } ^ { \left( \frac { \sqrt { x _ { n } } } { 4 } \right) }$$

with $x _ { 1 } = 0.5$ to find the values of $x _ { 2 }$ and $x _ { 3 }$, giving your answers to three decimal places.
\item Figure 1, on the page 3, shows a sketch of parts of the graphs of $y = \mathrm { e } ^ { \left( \frac { \sqrt { x } } { 4 } \right) }$ and $y = x$, and the position of $x _ { 1 }$.

On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of $x _ { 2 }$ and $x _ { 3 }$ on the $x$-axis.

\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
  \includegraphics[alt={},max width=\textwidth]{d3c66c34-b09c-4223-8383-cf0a68419bf9-3_1285_1543_356_296}
\end{center}
\end{figure}
\end{enumerate}

\hfill \mbox{\textit{AQA C3 2012 Q2 [7]}}