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Question
Find the value of x, such that f(x) is increasing function.
f(x) = 2x3 - 15x2 - 144x - 7
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Solution
f(x) = 2x3 - 15x2 - 144x - 7
∴ f'(x) = 6x2 - 30x - 144
f(x) is an increasing function, if f'(x) > 0
∴ 6(x2 - 5x - 24) > 0
∴ 6(x + 3)(x - 8) > 0
∴ (x + 3)(x - 8) > 0
ab > 0 ⇔ a > 0 and b > 0 or a < 0 or b < 0
∴ Either (x + 3) > 0 and (x – 8) > 0 or
(x + 3) < 0 and (x – 8) < 0
Case 1: x + 3 > 0 and x - 8 > 0
∴ x > -3 and x > 8
∴ x > 8
Case 2: x + 3 < 0 and x - 8 < 0
∴ x < - 3 or x < 8
∴ x < - 3
Thus, f(x) is an increasing function for x < -3, or x > 8 i.e., (-∞, - 3) ∪ (8, ∞).
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