The diagram displays the curve $y = x^3 \cos 2x$ for $0 \leq x \leq \tfrac{1}{4}\pi$. This curve reaches a maximum point at $M$, where $x = p$.
(a)[3]
Show that $p$ obeys the equation $p = \dfrac{1}{2}\tan^{-1}\!\left(\dfrac{3}{2p}\right)$.
(b)[2]
Show, by calculation, that $0.5 \le p \le 0.7$.
(c)[3]
Use the iterative formula based on the equation in part (a) to find $p$ correct to 3 decimal places. Record each iterate to 5 decimal places.
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