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\includegraphics[max width=\textwidth, alt={}, center]{8bdd1285-9e39-465a-8c09-bbe410504f9d-06_442_698_260_721} The diagram shows part of the curve with equation \(y = x ^ { 3 } \cos 2 x\). The curve has a maximum at the point \(M\).

1. Show that the \(x\)-coordinate of \(M\) satisfies the equation \(x = \sqrt [ 3 ] { 1.5 x ^ { 2 } \cot 2 x }\).
2. Use the equation in part (a) to show by calculation that the \(x\)-coordinate of \(M\) lies between 0.59 and 0.60.
3. Use an iterative formula, based on the equation in part (a), to find the \(x\)-coordinate of \(M\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.