5 The diagram shows part of a curve with a maximum point \(M\).
\includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-4_480_645_354_694}
The curve is defined for \(x \geqslant 0\) by the equation
$$y = 6 x - 2 x ^ { \frac { 3 } { 2 } }$$
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
(3 marks) - Hence find the coordinates of the maximum point \(M\).
- Write down the equation of the normal to the curve at \(M\).
- The point \(P \left( \frac { 9 } { 4 } , \frac { 27 } { 4 } \right)\) lies on the curve.
- Find an equation of the normal to the curve at the point \(P\), giving your answer in the form \(a x + b y = c\), where \(a , b\) and \(c\) are positive integers.
- The normals to the curve at the points \(M\) and \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
\(6 \quad\) A curve \(C\), defined for \(0 \leqslant x \leqslant 2 \pi\) by the equation \(y = \sin x\), where \(x\) is in radians, is sketched below. The region bounded by the curve \(C\), the \(x\)-axis from 0 to 2 and the line \(x = 2\) is shaded.
\includegraphics[max width=\textwidth, alt={}, center]{258f0400-6e3b-406c-9f86-acc9fff4e094-5_441_789_466_612}