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The diagram shows the curve \(y = x \cos 2 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The point \(M\) is a maximum point.
- Show that the \(x\)-coordinate of \(M\) satisfies the equation \(1 = 2 x \tan 2 x\).
- The equation in part (i) can be rearranged in the form \(x = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x } \right)\). Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 2 } \tan ^ { - 1 } \left( \frac { 1 } { 2 x _ { n } } \right) ,$$
with initial value \(x _ { 1 } = 0.4\), to calculate the \(x\)-coordinate of \(M\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.
- Use integration by parts to find the exact area of the region enclosed between the curve and the \(x\)-axis from 0 to \(\frac { 1 } { 4 } \pi\).