Answer:
a) Domain: all real values of x
Range: all real values of y
b) Relative maxima: (0,3)
Relative minima: (2,-1)
c) yes, (1,1)
d) x < 0, x > 2
Step-by-step explanation:
a) Domain: all real values of x
Range: all real values of y
b) dy/dx = 3x² - 6x = 0
3x(x - 2) = 0
x = 0, y = 0³ - 3(0)² + 3 = 3
x = 2, y = 2³ - 3(2)² + 3 = -1
d²y/dx² = 6x - 6
x = 0, d²y/dx² = -6 < 0 (max)
x = 2, d²y/dx² = 6 > 0 (min)
c) d²y/dx² = 0 at points of infection
6x - 6 = 0
x = 1
y = 1³ - 3(1)² + 3 = 1
dy/dx at x = 0.9: 3(0.9)² - 6-(0.9)
= -0.298
dy/dx at x = 1.1: 3(1.1²) - 6(1.1)
= -0.297
Decreasing on both sides, therefore (1,1) is a point of inflection
d) increasing when dy/dx > 0
3x² - 6x > 0
Roots: x = 0, 2
x < 0, x > 2
