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JEE Main202622 January 2026Morning ShiftMathematicsDifferential EquationsActual

Let f:[1, ) R be a differentiable function. If 6 ₁^ x f(t) d t=3 x f(x)+x³-4 for all x 1 , then the value of f(2)-f(3) is

Options

  1. A-4
  2. B3
  3. C4
  4. D-3

Correct answer

B. 3

Step-by-step solution

Differentiating 6 ₁^x f(t)dt = 3xf(x) + x^3 - 4 : 6f(x) = 3f(x) + 3xf'(x) + 3x^2 f(x) - xf'(x) = x^2 . d dx ( f(x) x ) = xf'(x)-f(x) x^2 = -1 . Integrating: f(x) x = -x + C f(x) = -x^2 + Cx . At x = 1 : 0 = 3f(1) - 3 f(1) = 1 . So -1 + C = 1 C = 2 . f(x) = -x^2 + 2x . f(2) = 0 , f(3) = -3 . f(2) - f(3) = 3 .

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