9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{b822842d-ee62-40ce-a8de-967e556a80a8-26_915_912_255_580}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 is a sketch of the curve \(C\) with equation
$$y = 2 x ^ { \frac { 3 } { 2 } } ( 4 - x ) \quad x \geqslant 0$$
The point \(P\) is the stationary point of \(C\).
- Find, using calculus, the \(x\) coordinate of \(P\).
The region \(R _ { 1 }\), shown shaded in Figure 1, is bounded by \(C\) and the \(x\)-axis.
The region \(R _ { 2 }\), also shown shaded in Figure 1, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = k\), where \(k\) is a constant.
Given that the area of \(R _ { 1 }\) is equal to the area of \(R _ { 2 }\) - find, using calculus, the exact value of \(k\).