Find the intervals in which the following functions are strictly increasing or strictly decreasing:
2x³ – 15x² + 36x + 6

Answer

Let f (x) = 2 x³ – 15x² + 36x + 6
f ' (x) = 6x² – 30x + 36 = 6 (x² – 5 x + 6) = 6 (x – 2) (x – 3)
(a) For f (x) to be increasing, f'(x) > 0
i.e., 6 (x – 2) (x – 3) > 0 or (x – 2) (x – 3) > 0
⇒ either x < 2 or x > 3
∴ f (x) is increasing in x < 2 or x > 3.
(b) For f (x) to be decreasing, f ' (x) < 0
i.e. 6 (x – 2) (x – 3) < 0 or (x – 2) (x – 3) < 0 ⇒ 2 < x < 3
∴ f (x) is decreasing in 2 < x < 3