The interval on which the function f(x) = 2x3 + 9x2 + 12x – 1 is decreasing is ______.

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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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