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COMEDK2024Evening ShiftMathematicsApplication of DerivativesActual

The function f(x)= ⁻¹( x+ x) is an increasing function in

Options

  1. A(- 2 , 2 )
  2. B( 4 , 2 )
  3. C(- 2 , 4 )
  4. D(0, 2 )

Correct answer

C. (- 2 , 4 )

Step-by-step solution

Given f(x) = ⁻¹( x + x) . To determine the interval where f(x) is increasing, we find the derivative f'(x) and set f'(x) > 0 . f'(x) = 1 1 + ( x + x)^2 d dx ( x + x) f'(x) = x - x 1 + ( ^2 x + ^2 x + 2 x x) f'(x) = x - x 1 + 1 + (2x) = x - x 2 + (2x) Since the denominator 2 + (2x) is always positive for all real x , the sign of f'(x) depends only on the numerator x - x . For f(x) to be increasing, we require f'(x) > 0 , which implies x - x > 0 . x > x Dividing by x (assuming x > 0 , which holds in the interval (- /

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