4 The equation of a curve is \(y = 9 x ^ { 2 } - x ^ { 4 }\).
- Show that the curve meets the \(x\)-axis at the origin and at \(x = \pm a\), stating the value of \(a\).
- Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
Hence show that the origin is a minimum point on the curve. Find the \(x\)-coordinates of the maximum points.
- Use calculus to find the area of the region bounded by the curve and the \(x\)-axis between \(x = 0\) and \(x = a\), using the value you found for \(a\) in part (i).