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Question
Find the intervals in which the function `f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12` is (a) strictly increasing, (b) strictly decreasing
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Solution
We have
`f(x) = x^4/4 - x^3 - 5x^2 + 24x + 12`
`=> f'(x) = x^3 -3x^2 - 10x + 24`
As x = 2 satisfies the above equation. Therefore, (x − 2) is a factor.
On performing long division
`(x^3 - 3x^2- 10x + 24)/(x -2) = x^2 -x -12 = (x + 3)(x+4)`
`=> f'(x) = (x - 2) (x + 3) (x -4)`
Here, the critical points are 2, −3, and 4.
The possible intervals are (−∞, −3), (−3, 2), (2, 4), (4, ∞)
a) For f(x) to be strictly increasing, we must have
f'(x) > 0
`=> (x - 2)(x + 3) (x - 4) > 0`
`=> x in (-3,2) U (4,oo)`
So, f(x) is strictly increasing on x∈(−3, 2) ∪ (4, ∞).
b) For f(x) to be strictly decreasing, we must have
f'(x) < 0
`=> (x - 2)(x+3)(x - 4) < 0`
`=> x in (-oo, -3) U (2,4)``
So, f(x) is strictly decreasing on x∈(−∞, −3) ∪ (2, 4).
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