A cubic curve has equation y x3 3x2 1. (i) Use calculus to find the coordinates of the turning points on this curve. Determine the nature of these turning points.

y′=3x2 −6x

use of y′ = 0

0= 3x^2 - 6x

0= 3x(x-2)

therefore either x=0 or x=2

when x=0 y=1, when x=2 y=-3

(0, 1) or (2, −3) 

y''=6x-6

when x=0 y''=6(0)-6 = -6

as y'' is negative this means at x=o the curve is at a maximum

whgen x=2 y''=6(2)-6=6

as y''is positive, this means at x=2 the curve is at a maximum