- In this question you must show detailed reasoning.
Solutions relying entirely on calculator technology are not acceptable.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{6f926d53-c6de-4eb7-9d18-596f61ec26e1-26_723_455_413_804}
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\caption{Figure 3}
\end{figure}
Figure 3 shows
- the curve \(C _ { 1 }\) with equation \(y = x ^ { 3 } - 5 x ^ { 2 } + 3 x + 14\)
- the circle \(C _ { 2 }\) with centre \(T\)
The point \(T\) is the minimum turning point of \(C _ { 1 }\)
Using Figure 3 and calculus,
- find the coordinates of \(T\)
The curve \(C _ { 1 }\) intersects the circle \(C _ { 2 }\) at the point \(A\) with \(x\) coordinate 2
- Find an equation of the circle \(C _ { 2 }\)
The line \(l\) shown in Figure 3, is the tangent to circle \(C _ { 2 }\) at \(A\)
- Show that an equation of \(l\) is
$$y = \frac { 1 } { 3 } x + \frac { 22 } { 3 }$$
The region \(R\), shown shaded in Figure 3, is bounded by \(C _ { 1 } , l\) and the \(y\)-axis.
- Find the exact area of \(R\).