12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b85872d4-00b2-499b-9765-f7559d3de66a-19_1011_1349_237_310}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of part of the curve \(C\) with equation \(y = x ^ { 2 } - \frac { 1 } { 3 } x ^ { 3 } C\) touches the \(x\)-axis at the origin and cuts the \(x\)-axis at the point \(A\).
- Show that the coordinates of \(A\) are \(( 3,0 )\).
- Show that the equation of the tangent to \(C\) at the point \(A\) is \(y = - 3 x + 9\)
The tangent to \(C\) at \(A\) meets \(C\) again at the point \(B\), as shown in Figure 5.
- Use algebra to find the \(x\) coordinate of \(B\).
The region \(R\), shown shaded in Figure 5, is bounded by the curve \(C\) and the tangent to \(C\) at \(A\).
- Find, by using calculus, the area of region \(R\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)