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JEE Main20265 April 2026Evening ShiftMathematicsLimitsActual

Let f(x) = _ y 0 (1 - (xy)) (xy) y^3 . Then the number of solutions of the equation f(x) = x , x R is :

Options

  1. A0
  2. B2
  3. C3
  4. D1

Correct answer

C. 3

Step-by-step solution

The given function is f(x) = _ y 0 (1 - (xy)) (xy) y^3 . Multiplying and dividing by x^3 , we get: f(x) = _ y 0 1 - (xy) (xy)^2 (xy) xy x^3 Using the standard limits _ t 0 1 - t t^2 = 1 2 and _ t 0 t t = 1 , we obtain: f(x) = 1 2 1 x^3 = x^3 2 We need to find the number of solutions for the equation f(x) = x , which is x^3 2 = x . Let g(x) = x^3 2 - x . Differentiating with respect to x : g'(x) = 3x^2 2 - x g''(x) = 3x + x For x > 0 , g''(x) > 0 , which implies that g'(x) is strictly increasing on (0, ) . We have g

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