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JEE Main202622 January 2026Evening ShiftMathematicsDefinite IntegrationActual

Let f(x)=[x]²-[x+3]-3, x R , where [] is the greatest integer funtion. Then

Options

  1. A₀² f(x) d x=-6
  2. Bf(x)<0 only for x [-1,3)
  3. Cf(x)>0 only for x [4, )
  4. Df(x)=0 for finitely many values of x

Correct answer

B. f(x)<0 only for x [-1,3)

Step-by-step solution

For x [n, n+1) where n is an integer, [x] = n and [x+3] = n+3 . Thus f(x) = n^2 - (n+3) - 3 = n^2 - n - 6 = (n-3)(n+2) . For f(x) This corresponds to x [-1, 3) . We can verify: For n = -1 : f = 1 - 2 - 3 = -4 For n = 0,1 : f = -6 For n = 2 : f = 4 - 5 - 3 = -4 For n = 3 : f = 9 - 6 - 3 = 0 For n 4 , (n-3)(n+2) > 0 . Therefore f(x) < 0 only for x [-1, 3) .

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