The curve \(C\) has equation \[y=x(x+1)(x-2)^4.\] Show that the gradient of \(C\) is \((x-2)^3(6x^2+x-2)\) and find the coordinates of all the stationary points. Determine the nature of each stationary point and sketch \(C\).

In separate diagrams draw sketches of the curves whose equations are:

  1. \(y^2=x(x+1)(x-2)^4\);

  2. \(y=x^2(x^2+1)(x^2-2)^4\).

In each case, you should pay particular attention to the points where the curve meets the \(x\) axis.