Question
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
6 + 12x + 3x2 – 2x3
Solution
Let f (x) = 6 + 12x + 3x2 – 2x3 = – 2x3 + 3x2 + 12x + 6
∴ f ' (x) = – 6x2 + 6x + 12 = – 6 (x2 – x – 2) = – 6 (x + 1 ) (x – 2)
(a) For f (x) to be increasing, f ' (x) > 0
or – 6 (x + 1) (x – 2) > 0 or (x + 1) (x – 2) < 0
⇒ – 1 < x < 2
∴ f (x) is increasing for –1 < x < 2
(b) For f (x) to be decreasing , f ' (x) < 0
or – 6 (x + 1) (x – 2) < 0 or (x + 1) (– 2) > 0
⇒ either x < – 1 or x > 2
∴ f (x) is decreasing for x < – 1 or x > 2.
