15.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b99072a-cd16-4c1d-9e44-085926a3ba24-24_591_570_255_678}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of part of the curve \(C\) with equation
$$y = x ^ { 3 } + 10 x ^ { \frac { 3 } { 2 } } + k x , \quad x \geqslant 0$$
where \(k\) is a constant.
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
The point \(P\) on the curve \(C\) is a minimum turning point.
Given that the \(x\) coordinate of \(P\) is 4 - show that \(k = - 78\)
The line through \(P\) parallel to the \(x\)-axis cuts the \(y\)-axis at the point \(N\).
The finite region \(R\), shown shaded in Figure 5, is bounded by \(C\), the \(y\)-axis and \(P N\). - Use integration to find the area of \(R\).