Determining Intervals on Which a Function is Increasing or Decreasing

Question

Determine where the function f(x)=(x-2)(x-3)² is increasing or decreasing. Verify your solution graphically.  Show all steps. Submit your answer in the submission box below.

Answer 1

f(x) = (x - 2)(x² - 6x + 9) = x³ - 6x² + 9x - 2x² + 12x - 18 = x³ - 8x²+ 21x - 18

f'(x) = 3x² - 16x + 21 = (3x - 7)(x - 3)

f'(x) = 0 when x = 7/3 or x = 3

When x < 7/3, 3x - 7 and x - 3 are both negative. So, f'(x) > 0. Therefore, f is increasing

When 7/3 < x < 3, 3x - 7 is positive and x - 3 is negative. So, f'(x) < 0. Therefore, f is decreasing

When x > 3, 3x - 7 and x - 3 are both positive. So, f'(x) > 0. Therefore, f is increasing.