7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{1c700103-ecab-4a08-b411-3f445ed88885-22_707_1047_264_463}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation
$$y = 2 \cos 3 x - 3 x + 4 \quad x > 0$$
where \(x\) is measured in radians.
The curve crosses the \(x\)-axis at the point \(P\), as shown in Figure 3.
Given that the \(x\) coordinate of \(P\) is \(\alpha\),
- show that \(\alpha\) lies between 0.8 and 0.9
The iteration formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \arccos \left( 1.5 x _ { n } - 2 \right)$$
can be used to find an approximate value for \(\alpha\).
- Using this iteration formula with \(x _ { 1 } = 0.8\) find, to 4 decimal places, the value of
- \(X _ { 2 }\)
- \(X _ { 5 }\)
The point \(Q\) and the point \(R\) are local minimum points on the curve, as shown in Figure 3.
Given that the \(x\) coordinates of \(Q\) and \(R\) are \(\beta\) and \(\lambda\) respectively, and that they are the two smallest values of \(x\) at which local minima occur,
- find, using calculus, the exact value of \(\beta\) and the exact value of \(\lambda\).