12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{53865e15-3838-4551-b507-fe49549b87db-32_748_883_274_477}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Diagram not drawn to scale
Figure 2 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\), where
$$f ( x ) = \frac { x ^ { 3 } - 9 x ^ { 2 } - 81 x } { 27 }$$
The curve crosses the \(x\)-axis at the point \(A\), the point \(B\) and the origin \(O\). The curve has a maximum turning point at \(C\) and a minimum turning point at \(D\).
- Use algebra to find exact values for the \(x\) coordinates of the points \(A\) and \(B\).
- Use calculus to find the coordinates of the points \(C\) and \(D\).
The graph of \(y = \mathrm { f } ( x + a )\), where \(a\) is a constant, has its minimum turning point on the \(y\)-axis.
- Write down the value of \(a\).
\includegraphics[max width=\textwidth, alt={}, center]{53865e15-3838-4551-b507-fe49549b87db-35_29_37_182_1914}