Question
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
x3 – 6x2 – 36x +4
Solution
Let f (a) = x3 – 6 x2 – 36x + 4
f '(x) = 3x2 – 12x – 36 = 3 (x2 – 4x – 12) = 3 (x + 2) (x – 6)
(a) For f (x) to be increasing, f ' (x) > 0
i.e. 3 (x + 2) (x – 6) > 0 or (x + 2) (x – 6) > 0
∴ either x < – 2 or x > 6
∴ f (x) is increasing in x > 6 and x < – 2.
(b) For f (x) to be decreasing, f ' (x) < 0
i.e. 3 (x + 2) (x – 6) < 0 or (x + 2)(x – 6) < 0
⇒ – 2 < x < 6
∴ f (x) is decreasing in – 2 < x < 6.
