7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{5b698944-41ac-4072-b5e1-c580b7752c39-20_689_712_248_680}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = 2 x ( 1 + x ) \ln x , \quad x > 0$$
The curve has a minimum turning point at \(A\).
- Find f'(x)
- Hence show that the \(x\) coordinate of \(A\) is the solution of the equation
$$x = \mathrm { e } ^ { - \frac { 1 + x } { 1 + 2 x } }$$
- Use the iteration formula
$$x _ { n + 1 } = \mathrm { e } ^ { - \frac { 1 + x _ { n } } { 1 + 2 x _ { n } } } , \quad x _ { 0 } = 0.46$$
to find the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\) to 4 decimal places.
- Use your answer to part (c) to estimate the coordinates of \(A\) to 2 decimal places.