CBSE Class 12-science Answered

We have,

f (x) = x⁴ – 8x³ + 22x² – 24x + 21.

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

f ¢ (x) = 4(x – 1) (x² – 5x + 6)

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

f ¢ (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

f ¢ (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)