10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-28_655_869_255_541}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows a sketch of the curve \(C\) with equation
$$y = \frac { 2 } { 7 } x ^ { 3 } + \frac { 1 } { 7 } x ^ { 2 } - \frac { 5 } { 2 } x + k$$
where \(k\) is a constant.
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\)
The line \(l\), shown in Figure 5, is the normal to \(C\) at the point \(A\) with \(x\) coordinate \(- \frac { 7 } { 2 }\) Given that \(l\) is also a tangent to \(C\) at the point \(B\),
- show that the \(x\) coordinate of the point \(B\) is a solution of the equation
$$12 x ^ { 2 } + 4 x - 33 = 0$$
- Hence find the \(x\) coordinate of \(B\), justifying your answer.
Given that the \(y\) intercept of \(l\) is - 1
- find the value of \(k\).
\includegraphics[max width=\textwidth, alt={}]{3cf69966-e825-4ff0-a6e8-c5dfdc92c53f-32_2640_1840_118_114}