Question
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
f(x) = 2x3 – 21x2 + 36x – 40
Solution
Here f (x) = 2x 3 – 21x2 + 36x – 40
∴ f '(x) = 6x2 – 42x + 36 = 6(x2 – 7 x + 6) = 6 (x – 1) (x – 6)
∴ f '(x) = 0 gives us 6 (x – 1) (x – 6) = 0
∴ x = 1, 6
The points x = 1, 6 divide the real line into three intervals – (∞, 1), (1, 6), (6, ∞).
In the interval (– ∞, 1), f ' (x) > 0
∴ f (x) is strictly increasing in (– ∞, 1)
In the interval (1, 6), f '(x) < 0
∴ f (x) is strictly decreasing (1, 6)
In the interval (6, ∞), f ' (x) > 0
∴ f (x) is strictly increasing in (6, ∞)
∴ we see that f (x) is strictly increasing in (– ∞, 1) ∪ (6, ∞) and strictly decreasing in (1, 6).
