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Question
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.
Options
`[–1, oo)`
[– 2, – 1]
`(-oo, -2]`
[– 1, 1]
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Solution
The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is [– 2, – 1].
Explanation:
The given function is f(x) = 2x3 + 9x2 + 12x – 1
f'(x) = 6x2 + 18x + 12
For increasing and decreasing f'(x) = 0
∴ 6x2 + 18x + 12 = 0
⇒ x2 + 3x + 2 = 0
⇒ x2 + 2x + x + 2 = 0
⇒ x(x + 2) + 1(x + 2) = 0
⇒ (x + 2)(x + 1) = 0
⇒ x = – 2, x = – 1
The possible intervals are `(–oo, – 2), (– 2, – 1), (– 1, oo)`
Now f'(x) = (x + 2) (x + 1)
⇒ `"f'"(x)_((-oo"," -2))` = (–) (–) = (+) increasing
⇒ `"f'"(x)_((-2"," -1))` = (+) (–) = (–) decreasing
⇒ `"f'"(x)_((-1"," oo))` = (+) (+) = (+) increasing.
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