Question
Find the intervals in which the following function f(x) is
(a) increasing (b) decreasing:
f (x) = 2x3 – 9x2 + 12x + 15
Solution
Let f (x) = 2x3 – 9x2 + 12x + 15
∴ (x) = 6x2 – 18x + 12 = 6(x2 – 3x + 2) = 6 (x – 1) (x – 2)
f ' (x) = 0 gives us 6 (x – 1) (x – 2) = 0 ⇒ x = 1 , 2
The points x = 1, 2 divide the real line into three intervals (– ∞ , 1), (1, 2), (2, ∞)
1. In the interval (– ∞ , 1), f ' (x) > 0
∴ f (x) is increasing in (– ∞ , 1)
2. In the interval (1, 2), f ' (x) < 0
∴ f (x) is decreasing in (1, 2)
3. In the interval (2, ∞), f ' (x) > 0
∴ f (x) is increasing in (2, ∞)
