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Question
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval ______.
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Solution
The function f(x) = `(2x^2 - 1)/x^4`, x > 0, decreases in the interval `(1, oo)`.
Explanation:
We have f(x) = `(2x^2 - 1)/x^4`
f'(x) = `(x^4(4x) - (2x^2 - 1) * 4x^3)/x^8`
⇒ f'(x) = `(4x^5 - (2x^2 - 1) * 4x^3)/x^8`
= `(4x^3[x^2 - 2x^2 + 1])/x^8`
= `(4(-x^2 + 1))/x^5`
For decreasing the function f'(x) < 0
∴ `(4(-x^2 + 1))/x^5 < 0`
⇒ `-x^2 + 1 < 0`
⇒ x2 < 1
∴ x > ± 1
⇒ `x ∈ (1, oo)`
Hence, the required interval is `(1, oo)`
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