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MHT CET202520 Apr 2025Evening ShiftMathematicsApplication of DerivativesActual

If 2 f (x)+3 f ( 1 x )=x^2+1, x 0 and y =5 x^2 f (x) , then y is strictly increasing in

Options

  1. A(0, 1 2 )
  2. B( -2 5 , 0 )
  3. C( 1 2 , 5 2 )
  4. D( -1 2 , 0 )

Correct answer

A. (0, 1 2 )

Step-by-step solution

The function y is strictly increasing when its derivative dy dx > 0 . From the equation 2f(x) + 3f( 1 x ) = x^2 + 1 , replacing x with 1 x yields 2f( 1 x ) + 3f(x) = 1 x^2 + 1 . Solving this system gives f(x) = 1 5 ( 3 x^2 + 1 - 2x^2) . Substituting into y = 5x^2 f(x) yields y = 3 + x^2 - 2x^4 . Differentiating gives dy dx = 2x - 8x^3 = 2x(1 - 4x^2) = 2x(1 - 2x)(1 + 2x) . The critical points are x = - 1 2 , x = 0 , and x = 1 2 . Testing intervals shows dy dx > 0 on (- , - 1 2 ) and (0, 1 2 ) . Among the options, on

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