3.83: Cubic f(x)=(2x+1)(3x−2)2f(x) = (2x + 1)(3x - 2)^{2}: calculus and sketch

0606/11/O/N/23 — Question 7 · 10 marks

A curve has equation y=f(x)y = f(x), where f(x)=(2x+1)(3x−2)2f(x) = (2x + 1)(3x - 2)^{2}.
(a)
Show that f′(x)f'(x) can be written in the form 2(3x−2)(px+q)2(3x - 2)(px + q), where pp and qq are integers.
[3]
(b)
Hence find the coordinates of the stationary points on the curve.
[2]
(c)
On the axes below, sketch the graph of y=f(x)y = f(x), stating the intercepts with the coordinate axes.
[3]
(d)
Find the values of kk such that the equation f(x)=kf(x) = k has 3 distinct solutions.
[2]