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\includegraphics[max width=\textwidth, alt={}, center]{da5162f3-b5d5-417f-9b6c-5ae0024f22d9-10_542_789_260_676}
The diagram shows the curve
$$y = x ^ { 2 } + 3 x + 1 + 5 \cos \frac { 1 } { 2 } x .$$
The curve crosses the \(y\)-axis at the point \(P\) and the gradient of the curve at \(P\) is \(m\). The point \(Q\) on the curve has \(x\)-coordinate \(q\) and the gradient of the curve at \(Q\) is \(- m\).
- Find the value of \(m\) and hence show that \(q\) satisfies the equation
$$x = a \sin \frac { 1 } { 2 } x + b ,$$
where the values of the constants \(a\) and \(b\) are to be determined.
- Show by calculation that \(- 4.5 < q < - 4.0\).
- Use an iterative formula based on the equation in part (i) to find the value of \(q\) correct to 3 significant figures. Give the result of each iteration to 5 significant figures.