7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a80a71cb-42e0-4587-8f8e-bacd69b8d07a-11_481_858_228_552}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = \left( x ^ { 2 } + 3 x + 1 \right) \mathrm { e } ^ { x ^ { 2 } }$$
The curve cuts the \(x\)-axis at points \(A\) and \(B\) as shown in Figure 2 .
- Calculate the \(x\) coordinate of \(A\) and the \(x\) coordinate of \(B\), giving your answers to 3 decimal places.
- Find \(\mathrm { f } ^ { \prime } ( x )\).
The curve has a minimum turning point at the point \(P\) as shown in Figure 2.
- Show that the \(x\) coordinate of \(P\) is the solution of
$$x = - \frac { 3 \left( 2 x ^ { 2 } + 1 \right) } { 2 \left( x ^ { 2 } + 2 \right) }$$
- Use the iteration formula
$$x _ { n + 1 } = - \frac { 3 \left( 2 x _ { n } ^ { 2 } + 1 \right) } { 2 \left( x _ { n } ^ { 2 } + 2 \right) } , \quad \text { with } x _ { 0 } = - 2.4$$
to calculate the values of \(x _ { 1 } , x _ { 2 }\) and \(x _ { 3 }\), giving your answers to 3 decimal places.
The \(x\) coordinate of \(P\) is \(\alpha\).
- By choosing a suitable interval, prove that \(\alpha = - 2.43\) to 2 decimal places.