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\includegraphics[max width=\textwidth, alt={}, center]{72042f09-3495-42e9-bee9-96ec5ac0bf0c-06_352_643_274_744}
The diagram shows the part of the curve \(y = x ^ { 2 } \cos 3 x\) for \(0 \leqslant x \leqslant \frac { 1 } { 6 } \pi\), and its maximum point \(M\), where \(x = a\).
- Show that \(a\) satisfies the equation \(a = \frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { 2 } { 3 a } \right)\).
- Use an iterative formula based on the equation in (a) to determine \(a\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.