6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{16c69ee4-255e-4d77-955a-92e1eb2f7d3e-09_458_1164_239_383}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation
$$y = 2 \cos \left( \frac { 1 } { 2 } x ^ { 2 } \right) + x ^ { 3 } - 3 x - 2$$
The curve crosses the \(x\)-axis at the point \(Q\) and has a minimum turning point at \(R\).
- Show that the \(x\) coordinate of \(Q\) lies between 2.1 and 2.2
- Show that the \(x\) coordinate of \(R\) is a solution of the equation
$$x = \sqrt { 1 + \frac { 2 } { 3 } x \sin \left( \frac { 1 } { 2 } x ^ { 2 } \right) }$$
Using the iterative formula
$$x _ { n + 1 } = \sqrt { 1 + \frac { 2 } { 3 } x _ { n } \sin \left( \frac { 1 } { 2 } x _ { n } ^ { 2 } \right) } , \quad x _ { 0 } = 1.3$$
- find the values of \(x _ { 1 }\) and \(x _ { 2 }\) to 3 decimal places.