16.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{aa75f1c1-ee97-4fee-af98-957e6a3fbba1-25_739_1308_278_328}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{figure}
Figure 6 shows a sketch of part of the curve \(C\) with equation
$$y = x ( x - 1 ) ( x - 2 )$$
The point \(P\) lies on \(C\) and has \(x\) coordinate \(\frac { 1 } { 2 }\)
The line \(l\), as shown on Figure 6, is the tangent to \(C\) at \(P\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
- Use part (a) to find an equation for \(l\) in the form \(a x + b y = c\), where \(a\), \(b\) and \(c\) are integers.
The finite region \(R\), shown shaded in Figure 6, is bounded by the line \(l\), the curve \(C\) and the \(x\)-axis.
The line \(l\) meets the curve again at the point \(( 2,0 )\)
- Use integration to find the exact area of the shaded region \(R\).