6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fc5d0d07-b750-4646-bdcb-419a290200c9-4_433_1011_221_529}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve with equation \(y = ( x + a ) ( x - b ) ^ { 2 }\), where \(a\) and \(b\) are positive constants. The curve cuts the \(x\)-axis at \(P\) and has a maximum point at \(S\) and a minimum point at \(Q\).
- Write down the coordinates of \(P\) and \(Q\) in terms of \(a\) and \(b\).
- Show that \(G\), the area of the shaded region between the curve \(P S Q\) and the \(x\)-axis, is given by \(G = \frac { ( a + b ) ^ { 4 } } { 12 }\).
The rectangle \(P Q R S T\) has \(R S T\) parallel to \(Q P\) and both \(P T\) and \(Q R\) are parallel to the \(y\)-axis.
- Show that \(\frac { G } { \text { Area of } P Q R S T } = k\), where \(k\) is a constant independent of \(a\) and \(b\) and find the value of \(k\).