6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{824d73c5-525c-4876-ad66-33c8f1664277-12_634_741_251_662}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = 8 \sin \left( \frac { 1 } { 2 } x \right) - 3 x + 9 \quad x > 0$$
and \(x\) is measured in radians.
The point \(P\), shown in Figure 2, is a local maximum point on the curve.
Using calculus and the sketch in Figure 2,
- find the \(x\) coordinate of \(P\), giving your answer to 3 significant figures.
The curve crosses the \(x\)-axis at \(x = \alpha\), as shown in Figure 2 .
Given that, to 3 decimal places, \(f ( 4 ) = 4.274\) and \(f ( 5 ) = - 1.212\) - explain why \(\alpha\) must lie in the interval \([ 4,5 ]\)
- Taking \(x _ { 0 } = 5\) as a first approximation to \(\alpha\), apply the Newton-Raphson method once to \(\mathrm { f } ( x )\) to obtain a second approximation to \(\alpha\).
Show your method and give your answer to 3 significant figures.