Determine for which values of x, the function f (x) = x⁴ – 2x² is increasing or decreasing.

f (x) = x⁴ – 2x²
∴    f '(x) = 4 x³ – 4x = 4x (x² – 1 )
(i) For f (x) to be increasing , f ' (x) > 0
Case I. Let x > 0 .
∴  f ' (x) > 0 when x² – 1 > 0 i.e.. (x – 1) (x + 1) > 0
∴  x does not lie in (– 1, 1)
Also x > 0
∴  we have x > 1
Including end point, f (x) is increasing in x ≥ 1.
Case II. Let x < 0
∴  (x) > 0 when x² – 1 < 0
i.e., when (x – 1) (x + 1) < 0
i.e., when x lies in (– 1, 1)
But x < 0
∴ we have – 1 < x < 0
∴  f (x) is increasing in – 1 < x < 0 or (– 1, 0)
(ii) For f (x) to be decreasing, f '(x) < 0
Case I. Let x > 0
∴  f ' (x) < 0 when x² – 1 < 0
ie., (x – 1) (x + 1) < 0 i.e., x lies in (– 1, 1)
But x > 0
∴  we have 0 < x < 1
∴  f (x) is decreasing in ( 0, 1)
Case II. Let x < 0
∴  f ' (x) < 0 when x² – 1 > 0
i.e. (x – 1) (x + 1) > 0
i.e., x does not lie in (– 1, 1)
Also x < 0
∴  we have x < – 1
∴ f (x) is decreasing in x < – 1.