9.
\includegraphics[max width=\textwidth, alt={}]{dc0ac5df-24a7-41b5-8410-f0e9b332ba64-22_602_752_246_657}
\section*{Figure 2}
Figure 2 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = 4 \left( x ^ { 2 } - 2 \right) \mathrm { e } ^ { - 2 x } \quad x \in \mathbb { R }$$
- Show that \(\mathrm { f } ^ { \prime } ( x ) = 8 \left( 2 + x - x ^ { 2 } \right) \mathrm { e } ^ { - 2 x }\)
- Hence find, in simplest form, the exact coordinates of the stationary points of \(C\).
The function g and the function h are defined by
$$\begin{array} { l l }
\mathrm { g } ( x ) = 2 \mathrm { f } ( x ) & x \in \mathbb { R }
\mathrm {~h} ( x ) = 2 \mathrm { f } ( x ) - 3 & x \geqslant 0
\end{array}$$ - Find (i) the range of \(g\)
(ii) the range of h