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Question
Find the values of x such that f(x) = 2x3 – 15x2 – 144x – 7 is decreasing function
Solution
f(x) = 2x3 – 15x2 – 144x – 7
∴ f'(x) = 6x2 – 30x – 144
= 6(x2 – 5x – 24)
= 6(x + 3)(x – 8)
f(x) is a decreasing function, if f'(x) < 0
6(x + 3)(x – 8) < 0
∴ (x + 3)(x – 8) < 0
ab < 0 ⇔ a > 0 and b < 0 or a < 0 and 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
Case 2: x + 3 < 0 and x – 8 > 0
∴ x < – 3 and x > 8, which is not possible
Thus, f(x) is a decreasing function for – 3 < x < 8 ,i.e., (– 3, 8).
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