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) = □ ∴ f'(x) = 6(□)( - Mathematics and Statistics

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Sum

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).

  Is there an error in this question or solution?
Chapter 1.4: Applications of Derivatives - Q.6

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