8. In this question you must show all stages of your working.
Solutions relying entirely on calculator technology are not acceptable.
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\includegraphics[alt={},max width=\textwidth]{124ee19f-8a49-42df-9f4b-5a1cc2139be9-24_739_736_411_605}
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\caption{Figure 2}
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Figure 2 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { 4 } { 3 } x ^ { 3 } - 11 x ^ { 2 } + k x \quad \text { where } k \text { is a constant }$$
The point \(M\) is the maximum turning point of \(C\) and is shown in Figure 2.
Given that the \(x\) coordinate of \(M\) is 2
- show that \(k = 28\)
- Determine the range of values of \(x\) for which \(y\) is increasing.
The line \(l\) passes through \(M\) and is parallel to the \(x\)-axis.
The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(y\)-axis. - Find, by algebraic integration, the exact area of \(R\).