6 Fig. 8 shows part of the curve \(y = x \cos 3 x\).
The curve crosses the \(x\)-axis at \(\mathrm { O } , \mathrm { P }\) and Q .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{11877196-83d9-4283-9eef-e617bea50c63-3_553_1178_622_529}
\captionsetup{labelformat=empty}
\caption{Fig. 8}
\end{figure}
- Find the exact coordinates of P and Q .
- Find the exact gradient of the curve at the point P .
Show also that the turning points of the curve occur when \(x \tan 3 x = \frac { 1 } { 3 }\).
- Find the area of the region enclosed by the curve and the \(x\)-axis between O and P , giving your answer in exact form.