9. The gradient, \(\frac { \mathrm { d } y } { \mathrm {~d} x }\), at the point \(( x , y )\) on a curve is given by
$$\frac { \mathrm { d } y } { \mathrm {~d} x } = 54 + 27 x - 6 x ^ { 2 }$$
- Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
[0pt]
[2 marks] - The curve passes through the point \(P \left( - 1 \frac { 1 } { 2 } , 4 \right)\).
Verify that the curve has a minimum point at \(P\).
[0pt]
[2 marks]
- Show that at the points on the curve where \(y\) is decreasing
$$2 x ^ { 2 } - 9 x - 18 > 0$$
- Solve the inequality \(2 x ^ { 2 } - 9 x - 18 > 0\).