Question
Determine the values of x for which the function f(x) = 2x3 – 24x + 5 is increasing or decreasing.
Solution
f (x) = 2 x3 – 24 x + 5
⇒ f ' (x) = 6 x2 – 24 = 6 (x2 – 4) = 6 (x – 2) (x + 2)
(i) For f (x) to be increasing, f ' (x) > 0
∴ 6 (x – 2) (x + 2) > 0 i.e. (x + 2) (x + 2) > 0
∴ either x < – 2
or x > 2
f (x) is increasing for x > 2 or x < – 2
(ii) For f (x) to be decreasing. f ' (x) < 0
∴ 6 (x – 2) (x + 2) < 0 or (x – 2) (x + 2) < 0
⇒ – 2 < x < 2
∴ f (x) is decreasing when – 2 < x < 2.
