Question
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
4x3 – 6x2 – 72x + 30
Solution
Here f (x) = 4x3 – 6x2 – 72x + 30
∴ f ' (x) = 12x2 – 12x – 72 = 12 (x2 – x – 6) = 12 (x – 3) (x + 2)
f '(x) = 0 gives us 12 (x – 3) (x + 2) = 0
∴ x = – 2, 3
The points x = – 2, 3 divide the real line into three disjoint intervals (– ∞ – 2), (–2, 3), (3, ∞).
Interval Sign of f' (x) Nature of function f
(–∞, –2) (–) (–) > 0 f is strictly increasing
(– 2, 3) (–) (+) < 0 f is strictly decreasing
(3, ∞) (+) (+) > 0 f is strictly increasing
(a) f is strictly increasing in the intervals (– ∞, – 2) and (3,
