10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{03548211-79cb-4629-b6ca-aa9dfcc77a33-17_598_736_223_603}
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\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { x ^ { 2 } \ln x } { 3 } - 2 x + 4 , \quad x > 0$$
Point \(A\) is the minimum turning point on the curve.
- Show, by using calculus, that the \(x\) coordinate of point \(A\) is a solution of
$$x = \frac { 6 } { 1 + \ln \left( x ^ { 2 } \right) }$$
- Starting with \(x _ { 0 } = 2.27\), use the iteration
$$x _ { n + 1 } = \frac { 6 } { 1 + \ln \left( x _ { n } ^ { 2 } \right) }$$
to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
- Use your answer to part (b) to deduce the coordinates of point \(A\) to one decimal place.