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Answer :
The interval where y increases for the function f(x) = (4x² - 1)/(x² + 1) is (-∞, -0.5) U (0.5, ∞) is 0.5-(-∞) = ∞.
To find the intervals where the function f(x) = (4x² - 1)/(x² + 1) increases, we need to find its derivative and determine its sign. The derivative of f(x) can be found using the quotient rule:
f'(x) = [(8x)(x² + 1) - (4x² - 1)(2x)]/(x² + 1)²
Simplifying this expression, we get:
f'(x) = (12x - 4x³)/(x² + 1)²
To find the critical points, we need to solve the equation f'(x) = 0:
12x - 4x³ = 0
4x(3 - x²) = 0
This gives us the critical points x = 0 and x = ±√3. We can now test the intervals between these critical points to determine the sign of f'(x) in each interval.
Testing x < -√3, we choose x = -4, and we get f'(-4) = (-224)/(17²) < 0. Therefore, f(x) is decreasing on this interval.
Testing -√3 < x < 0, we choose x = -1, and we get f'(-1) = (16)/(2²) > 0. Therefore, f(x) is increasing on this interval.
Testing 0 < x < √3, we choose x = 1, and we get f'(1) = (16)/(2²) > 0. Therefore, f(x) is increasing on this interval.
Testing x > √3, we choose x = 4, and we get f'(4) = (-224)/(17²) < 0. Therefore, f(x) is decreasing on this interval.
Hence, the interval where f(x) increases is (-∞, -0.5) U (0.5, ∞). Therefore, the answer is 0.5 - (-∞) = ∞.
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