9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9b0b8db0-79fd-4ad5-88c9-737447d9f894-26_698_744_255_593}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = x \left( x ^ { 2 } - 4 \right) e ^ { - \frac { 1 } { 2 } x }$$
- Find \(f ^ { \prime } ( x )\).
The line \(l\) is the normal to the curve at \(O\) and meets the curve again at the point \(P\). The point \(P\) lies in the 3rd quadrant, as shown in Figure 3.
- Show that the \(x\) coordinate of \(P\) is a solution of the equation
$$x = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x } }$$
- Using the iterative formula
$$x _ { n + 1 } = - \frac { 1 } { 2 } \sqrt { 16 + \mathrm { e } ^ { \frac { 1 } { 2 } x _ { n } } } \quad \text { with } x _ { 1 } = - 2$$
find, to 4 decimal places,
- the value of \(x _ { 2 }\)
- the \(x\) coordinate of \(P\).