Determine the intervals in which the function f (x) = x4 – 8x3 + 22x2 – 24x + 21 is decreasing or increasing

Asked by Topperlearning User | 6th Aug, 2014, 10:17: AM

Expert Answer:

We have,

f (x) = x4 – 8x3 + 22x2 – 24x + 21.

¢ (x) = 4x3 – 24x2 + 44x – 24 = 4(x3 – 6x2 + 11 x – 6)

¢ (x) = 4(x – 1) (x2 – 5x + 6)

For f (x) to be increasing, we must have

¢ (x) > 0

Þ 4(x – 1) (x2 – 5x + 6) > 0

Þ (x – 1) (x2 – 5x + 6) > 0

Þ (x – 1) (x – 2)(x – 3) > 0         [ 4 > 0]

Þ 1 < x < 2 or 3 < x < ¥

Þ x Î (1, 2) È (3, ¥)

So, f (x) is increasing on (1, 2) È (3, ¥)

For f (x) to be decreasing, we must have

¢ (x) < 0

Þ 4(x – 1) (x2 – 5x + 6) > 0

Þ (x – 1)(x2 – 5x + 6) < 0           [ 4 > 0]

Þ (x – 1) (x – 2) (x – 3) < 0

Þ 2 < x < 3 or x < 1

Þ x Î (2, 3) È (–¥, 1)

So, f (x) is decreasing on (2, 3) È (–¥, 1)

Answered by  | 6th Aug, 2014, 12:17: PM