Question

Find the intervals on which each of the following functions is (a) increasing (b) decreasing.

f(x) = (x - 1)(x - 2)²

Answer 1

f(x) = (x - 1)(x - 2)² = x² - 4x+4 x - 1 

= x³ - 4x²+ 4x - x²+ 4x - 4

f(x) = x³ - 5x²+8x - 4

f’(x) = 3x² - 10x+8

f’(x) = 3x² - 6x - 4x+8

= 3x(x - 2) - 4(x - 2)

= (3x - 4)(x - 2)

Function f(x) is decreasing for x ∈ [4/3,2] and increasing in x ∈ (-∞,4/3) ∪ (2, ∞)