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Question
By completing the following activity, find the values of x such that f(x) = 2x3 – 15x2 – 84x – 7 is decreasing function.
Solution: f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = `square`
∴ f'(x) = 6`(square) (square)`
Since f(x) is decreasing function.
∴ f'(x) < 0
Case 1: `(square)` > 0 and (x + 2) < 0
∴ x ∈ `square`
Case 2: `(square)` < 0 and (x + 2) > 0
∴ x ∈ `square`
∴ f(x) is decreasing function if and only if x ∈ `square`
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Solution
f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = 6x2 – 30x – 84
= 6(x2 – 5x – 14)
∴ f'(x) = 6(x – 7)(x + 2)
Since f(x) is decreasing function.
∴ f'(x) < 0
∴ 6(x – 7)(x + 2) < 0
∴ (x – 7)(x + 2) < 0
Case 1: (x – 7) > 0 and (x + 2) < 0
∴ x > 7 and x < – 2
∴ x ∈ `bb(cancel0)` , which is not possible
Case 2: (x – 7) < 0 and (x + 2) > 0
∴ x < 7 and x > – 2
∴ x ∈ (– 2, 7)
∴ f(x) is decreasing function if and only if x ∈ (– 2, 7).
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