7 The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), of a curve \(C\) at the point \(( x , y )\) is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 20 x - 6 x ^ { 2 } - 16$$
- Show that \(y\) is increasing when \(3 x ^ { 2 } - 10 x + 8 < 0\).
- Solve the inequality \(3 x ^ { 2 } - 10 x + 8 < 0\).
- The curve \(C\) passes through the point \(P ( 2,3 )\).
- Verify that the tangent to the curve at \(P\) is parallel to the \(x\)-axis.
- The point \(Q ( 3 , - 1 )\) also lies on the curve. The normal to the curve at \(Q\) and the tangent to the curve at \(P\) intersect at the point \(R\). Find the coordinates of \(R\).
(7 marks)