Short Answer TypeFind the intervals in which the following functions are strictly increasing or strictly decreasing:
2x3 – 8x2 + 10x + 5
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
2x3 – 6x2 – 48x + 17
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
f (x) = 2x3 – 9x2 + 12x + 30
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
f (x) = 2x3 – 3x2 – 36x + 7
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
f(x) = 2x3 – 21x2 + 36x – 40
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
4x3 – 6x2 – 72x + 30
Find the intervals in which the following functions are strictly increasing or strictly decreasing:
– 2x3 – 9x2 – 12x + 1
Long Answer TypeDetermine 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.
, is
Switch
