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\includegraphics[max width=\textwidth, alt={}, center]{729aa2f6-2b62-445f-a2aa-a63b45cb6b64-3_604_971_262_587}
The diagram shows the curve \(y = x ^ { 2 } \cos x\), for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\), and its maximum point \(M\).
- Show by differentiation that the \(x\)-coordinate of \(M\) satisfies the equation
$$\tan x = \frac { 2 } { x }$$
- Verify by calculation that this equation has a root (in radians) between 1 and 1.2.
- Use the iterative formula \(x _ { n + 1 } = \tan ^ { - 1 } \left( \frac { 2 } { x _ { n } } \right)\) to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.