CBSE Class 12-science Answered

We have,

f (x) = 2x³ + 9x² + 12x + 20

\ f ' (x) = 6x² + 18x + 12 = 6(x² + 3x + 2)

(i)   for f (x) to be increasing, we must have

f ' (x) > 0

=> 6(x² + 3x + 2) > 0

=> (x² + 3x + 2) > 0        [ 6 > 0 and 6 (x² + 3x + 2) > 0 \ x² + 3x + 2 > 0]

=> (x + 1) (x + 2) > 0

=> x < -2 or x > -1

 Therefore, (-¥, -2) & (-1, ¥)

So, f (x) is increasing on (¥, -2) & (-1, ¥)

(ii)   for f (x) to be decreasing, we must have

f ' (x) < 0

=> 6(x² + 3x + 2) < 0     

=> x² + 3x + 2 < 0          [ 6 > 0 and 6(x² + 3x + 2) < 0 \ x² + 3x + 2 < 0]

=> (x + 1)(x + 2) < 0

-2 < x < -1

So, f (x) is decreasing on (-2, -1)