8 Fig. 8 shows part of the curve \(y = x \cos 2 x\), together with a point P at which the curve crosses the \(x\)-axis.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0ee3d87a-0d9e-4fa5-b8f5-8b28489e65b5-4_421_965_349_550}
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\caption{Fig. 8}
\end{figure}
- Find the exact coordinates of P .
- Show algebraically that \(x \cos 2 x\) is an odd function, and interpret this result graphically.
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
- Show that turning points occur on the curve for values of \(x\) which satisfy the equation \(x \tan 2 x = \frac { 1 } { 2 }\).
- Find the gradient of the curve at the origin.
Show that the second derivative of \(x \cos 2 x\) is zero when \(x = 0\).
- Evaluate \(\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } x \cos 2 x \mathrm {~d} x\), giving your answer in terms of \(\pi\). Interpret this result graphically.