7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{57ea7a94-6939-4c12-a6fd-01cd6af73310-12_632_873_294_532}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 2 shows a sketch of part of the curve with equation
$$\mathrm { g } ( x ) = x ^ { 2 } ( 1 - x ) \mathrm { e } ^ { - 2 x } , \quad x \geqslant 0$$
- Show that \(\mathrm { g } ^ { \prime } ( x ) = \mathrm { f } ( x ) \mathrm { e } ^ { - 2 x }\), where \(\mathrm { f } ( x )\) is a cubic function to be found.
- Hence find the range of g .
- State a reason why the function \(\mathrm { g } ^ { - 1 } ( x )\) does not exist.