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Question
Find the value of x, such that f(x) is decreasing function.
f(x) = 2x3 – 15x2 – 84x – 7
Solution
f(x) = 2x3 – 15x2 – 84x – 7
∴ f'(x) = 6x2 – 30x – 84
= 6(x2 – 5x – 14)
= 6(x2 – 7x + 2x – 14)
= 6(x – 7)(x + 2)
f(x) is an decreasing function, if f'(x) < 0
∴ 6(x – 7)(x + 2) < 0
∴ (x – 7)(x + 2) < 0
ab < 0 `⇔` a > 0 and b < 0 or a < 0 or b > 0
∴ Either (x – 7) > 0 and (x + 2) < 0 or
(x – 7) < 0 and (x + 2) > 0
Case 1: x – 7 > 0 and x + 2 < 0
∴ x > 7 and x < –2, which is not possible.
Case 2: x – 7 < 0 and x + 2 > 0
∴ x < 7 and x > –2
Thus, f(x) is an decreasing function for –2 < x < 7 i.e., (–2, 7)
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