The graph depicts the curve $y = \sin 3x \cos x$ for $0 \leq x \leq \frac{1}{2}\pi$, together with its minimum point $M$. The shaded region $R$ lies between the curve and the $x$-axis.
(i)[3]
Expand $\sin(3x + x)$ and $\sin(3x - x)$ to prove that $\sin 3x \cos x = \frac{1}{2}(\sin 4x + \sin 2x)$.
(ii)[4]
Using the result from part (i) and showing all the required working, determine the exact area of the region $R$.
(iii)[5]
Using the result from part (i), express $\frac{dy}{dx}$ in terms of $\cos 2x$ and hence determine the $x$-coordinate of $M$, giving your answer correct to 2 decimal places.
Worked solution & mark scheme
This 12-mark question has a full step-by-step worked solution and mark scheme. One marking point: “State the correct expansion of $\sin(3x+x)$ or $\sin(3x-x)$” …