10 The equation of a curve is \(y = 7 + 6 x - x ^ { 2 }\).
- Use calculus to find the coordinates of the turning point on this curve.
Find also the coordinates of the points of intersection of this curve with the axes, and sketch the curve.
- Find \(\int _ { 1 } ^ { 5 } \left( 7 + 6 x - x ^ { 2 } \right) \mathrm { d } x\), showing your working.
- The curve and the line \(y = 12\) intersect at ( 1,12 ) and ( 5,12 ). Using your answer to part (ii), find the area of the finite region between the curve and the line \(y = 12\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{15b8f97b-c058-409f-907f-cb0a6102abc4-5_643_1034_331_513}
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\caption{Fig. 11}
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The equation of the curve shown in Fig. 11 is \(y = x ^ { 3 } - 6 x + 2\).