5 A curve, drawn from the origin \(O\), crosses the \(x\)-axis at the point \(P ( 4,0 )\).
The normal to the curve at \(P\) meets the \(y\)-axis at the point \(Q\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{14c2acbb-5f3e-40e2-8b88-162341ab9f71-3_526_629_916_813} The curve, defined for \(x \geqslant 0\), has equation $$y = 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } }$$

1. 1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
      (3 marks)
   2. Show that the gradient of the curve at \(P ( 4,0 )\) is - 2 .
   3. Find an equation of the normal to the curve at \(P ( 4,0 )\).
   4. Find the \(y\)-coordinate of \(Q\) and hence find the area of triangle \(O P Q\).
   5. The curve has a maximum point \(M\). Find the \(x\)-coordinate of \(M\).
2. 1. Find \(\int \left( 4 x ^ { \frac { 1 } { 2 } } - x ^ { \frac { 3 } { 2 } } \right) \mathrm { d } x\).
   2. Find the total area of the region bounded by the curve and the lines \(P Q\) and \(Q O\).