5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{f0f328ed-3550-4b8d-8b80-016df8773b21-07_465_565_296_701}
\end{figure}
Figure 2 shows part of the curve with equation
$$y = ( 2 x - 1 ) \tan 2 x , \quad 0 \leqslant x < \frac { \pi } { 4 }$$
The curve has a minimum at the point \(P\). The \(x\)-coordinate of \(P\) is \(k\).
- Show that \(k\) satisfies the equation
$$4 k + \sin 4 k - 2 = 0$$
The iterative formula
$$x _ { n + 1 } = \frac { 1 } { 4 } \left( 2 - \sin 4 x _ { n } \right) , x _ { 0 } = 0.3$$
is used to find an approximate value for \(k\).
- Calculate the values of \(x _ { 1 } , x _ { 2 } , x _ { 3 }\) and \(x _ { 4 }\), giving your answers to 4 decimal places.
- Show that \(k = 0.277\), correct to 3 significant figures.