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Find the values of x such that f(x) = 2x^{3} – 15x^{2} – 144x – 7 is decreasing function

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#### Solution

f(x) = 2x^{3} – 15x^{2} – 144x – 7

∴ f'(x) = 6x^{2} – 30x – 144

= 6(x^{2} – 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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