f'(x)=(1-x)(2-x)

Question

determine the intervals on which f(x) is increasing or decreasing

 

 

 

 

 

Answer 1

f'(x) = (1-x)(2-x)

f''(x) = 2x - 3

 

The extreme points (maxima and minima) occur where f'(x) = 0, which occurs when x = 1 and x = 2.  At x = 1, f''(x) = 2(1)-3 = -1, so f(x) has a maximum at x=1.  f''(2) = 2(2)-3 = 1, so f(x) has a minimum at x = 2.  Hence:

• f(x) is increasing on the interval (-∞,1)
• f(x) is decreasing on the interval (1,2)
• f(x) is increasing on the interval (2,∞)

 

Note: f(x) = (1/3)x³ - (3/2)x² + 2x