Question
Find the intervals in which the following function is increasing or decreasing:
x3 – 6x2 + 9x + 15.
Solution
Let f (x) = x3- 6x2 + 9x + 15
∴ f '(x) – 3x2 – 12x + 9 = 3 (x2 – 4 x + 3) = 3 (x – 1) (x – 3)
f ' (x) = 0 gives us 3(x – 1) (x – 3) = 0
∴ x = 1, 3
The points x = 1, 3 divide the real line into three intervals (– ∞, 1), (1, 3), (3, ∞).
(1) In the interval (– ∞ , 1), f ' (x) > 0
∴ f (x) is increasing in (– ∞, 1)
(2) In the interval (1, 3), f ' (x) < 0
∴ f ' (x) is decreasing in (1, 3).
(3) In the interval (3, ∞), f ' (x) > 0
∴ f (x) is increasing in (3, ∞).
