15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{bb1becd5-96c1-426d-9b85-4bbc4a61af27-42_695_1450_251_246}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of part of the graph \(y = \mathrm { f } ( x )\), where
$$f ( x ) = \frac { ( x - 3 ) ^ { 2 } ( x + 4 ) } { 2 } , \quad x \in \mathbb { R }$$
The graph cuts the \(y\)-axis at the point \(P\) and meets the positive \(x\)-axis at the point \(R\), as shown in Figure 5.
- State the \(y\) coordinate of \(P\).
- State the \(x\) coordinate of \(R\).
The line segment \(P Q\) is parallel to the \(x\)-axis. Point \(Q\) lies on \(y = \mathrm { f } ( x ) , x > 0\)
- Use algebra to show that the \(x\) coordinate of \(Q\) satisfies the equation
$$x ^ { 2 } - 2 x - 15 = 0$$
- Use part (b) to find the coordinates of \(Q\).
The region \(S\), shown shaded in Figure 5, is bounded by the curve \(y = \mathrm { f } ( x )\) and the line segment \(P Q\).
- Use calculus to find the exact area of \(S\).