7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8bd0bc33-e69e-4e51-aae7-288810c5db07-6_643_1374_173_351}
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\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve \(C _ { 1 }\) with equation \(y = \mathrm { f } ( x )\) where
$$\mathrm { f } ( x ) = \frac { x } { 3 } + \frac { 12 } { x } \quad x \neq 0$$
The lines \(x = 0\) and \(y = \frac { x } { 3 }\) are asymptotes to \(C _ { 1 }\). The point \(A\) on \(C _ { 1 }\) is a minimum and the point \(B\) on \(C _ { 1 }\) is a maximum.
- Find the coordinates of \(A\) and \(B\).
There is a normal to \(C _ { 1 }\), which does not intersect \(C _ { 1 }\) a second time, that has equation \(x = k\), where \(k > 0\).
- Write down the value of \(k\).
The point \(P ( \alpha , \beta ) , \alpha > 0\) and \(\alpha \neq k\), lies on \(C _ { 1 }\). The normal to \(C _ { 1 }\) at \(P\) does not intersect \(C _ { 1 }\) a second time.
- Find the value of \(\alpha\), leaving your answer in simplified surd form.
- Find the equation of this normal.
The curve \(C _ { 2 }\) has equation \(y = | \mathrm { f } ( x ) |\)
- Sketch \(C _ { 2 }\) stating the coordinates of any turning points and the equations of any asymptotes.
The line with equation \(y = m x + 1\) does not touch or intersect \(C _ { 2 }\).
- Find the set of possible values for \(m\).