10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{f3272b4c-d8dc-4f32-add9-153de90f4d0a-30_661_743_210_603}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of the curve \(C\) with equation
$$x = y \mathrm { e } ^ { 2 y } \quad y \in \mathbb { R }$$
- Show that
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y } { x ( 1 + 2 y ) }$$
Given that the straight line with equation \(x = k\), where \(k\) is a constant, cuts \(C\) at exactly two points,
- find the range of possible values for \(k\).