The function f(x) = (2x^2 - 1)/x^4, x > 0 decreases in the interval _______.

Question

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The function f(x) = (2x² - 1)/x^4, x > 0 decreases in the interval _______.

Answer 1

Now applying the quotient rule of differentiation and the differentiation of the constant term is 0 we get

⇒ x²-1=0

⇒ x²=1

⇒ x=±1

The intervals formed by these two critical numbers are (-∞, -1), (-1, 0), (0, 1) and (1, ∞)

(i) in the interval (-∞, -1), f’(x)>0

∴ f(x) is increasing in (-∞,-1)

(ii) in the interval (-1, 0), f’(x)<0

∴ f(x) is decreasing in(-1,0)

(iii) in the interval (0, 1), f’(x)>0

∴ f(x) is increasing in (1, ∞)

(iii) in the interval (1, ∞), f’(x)<0

∴ f(x) is decreasing in (1, ∞)

Hence the function f(x) = (2x² - 1)/x⁴, x > 0 decreases in the interval (1, ∞).