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
