4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c6bde466-61ec-437d-a3b4-84511a98d788-08_510_783_260_584}
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\caption{Figure 1}
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Figure 1 shows a sketch of part of the curve with equation \(y = 8 x - x \mathrm { e } ^ { 3 x } , x \geqslant 0\) The curve meets the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).
- Find the exact \(x\) coordinate of \(A\), giving your answer in its simplest form.
The curve has a maximum turning point at the point \(M\).
- Show, by using calculus, that the \(x\) coordinate of \(M\) is a solution of
$$x = \frac { 1 } { 3 } \ln \left( \frac { 8 } { 1 + 3 x } \right)$$
- Use the iterative formula
$$x _ { n + 1 } = \frac { 1 } { 3 } \ln \left( \frac { 8 } { 1 + 3 x _ { n } } \right)$$
with \(x _ { 0 } = 0.4\) to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.