7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{08291ac1-bdd4-4241-8959-7c89318fa5eb-18_554_1129_248_468}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve \(C\) with equation \(y = \mathrm { f } ( x )\) where
$$f ( x ) = e ^ { - x ^ { 2 } } \left( 2 x ^ { 2 } - 3 \right) ^ { 2 }$$
- Find the range of f
- Show that
$$\mathrm { f } ^ { \prime } ( x ) = 2 x \left( 2 x ^ { 2 } - 3 \right) \mathrm { e } ^ { - x ^ { 2 } } \left( A - B x ^ { 2 } \right)$$
where \(A\) and \(B\) are constants to be found.
Given that the line \(y = k\), where \(k\) is a constant, \(k > 0\), intersects the curve at exactly two distinct points,
- find the exact range of values of \(k\)