10.
\begin{figure}[h]
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\caption{Figure 3}
\includegraphics[alt={},max width=\textwidth]{29c7baa1-6929-448a-a756-319ea75dffa7-14_636_956_285_513}
\end{figure}
Figure 3 shows a sketch of part of the curve with equation \(y = x ^ { 3 } - 8 x ^ { 2 } + 20 x\). The curve has stationary points \(A\) and \(B\).
- Use calculus to find the \(x\)-coordinates of \(A\) and \(B\).
- Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\), and hence verify that \(A\) is a maximum.
The line through \(B\) parallel to the \(y\)-axis meets the \(x\)-axis at the point \(N\).
The region \(R\), shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line from \(A\) to \(N\). - Find \(\int \left( x ^ { 3 } - 8 x ^ { 2 } + 20 x \right) \mathrm { d } x\).
- Hence calculate the exact area of \(R\).