7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b9472037-c143-4b68-86e2-801f71029773-20_554_559_264_753}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The curve \(C\) has equation \(y = \mathrm { f } ( x )\), where
$$\mathrm { f } ( x ) = x ^ { 3 } \sqrt { 4 x + 7 } \quad x \geqslant - \frac { 7 } { 4 }$$
- Show that
$$\mathrm { f } ^ { \prime } ( x ) = \frac { k x ^ { 2 } ( 2 x + 3 ) } { \sqrt { 4 x + 7 } }$$
where \(k\) is a constant to be found.
The point \(P\), shown in Figure 3, is the minimum turning point on \(C\).
- Find the coordinates of \(P\).
- Hence find the range of the function g defined by
$$g ( x ) = - 4 f ( x ) \quad x \geqslant - \frac { 7 } { 4 }$$
The point \(Q\) with coordinates \(\left( \frac { 1 } { 2 } , \frac { 3 } { 8 } \right)\) lies on \(C\).
- Find the coordinates of the point to which \(Q\) is mapped when \(C\) is transformed to the curve with equation
$$y = 40 \mathrm { f } \left( x - \frac { 3 } { 2 } \right) - 8$$