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MHT CET202526 Apr 2025Evening ShiftMathematicsDifferentiationActual

If f (x)=3 x^2+2 x f ^ (1)+ f ^ (2) , then f (x)= ______

Options

  1. A3 x^2+8 x+4
  2. B3 x^2+12 x+12
  3. C3 x^2-12 x+6
  4. D3 x^2-18 x+5

Correct answer

C. 3 x^2-12 x+6

Step-by-step solution

Given the function f (x)=3x^2+2x f ^ (1)+ f ^ (2) , the derivative terms are constants since they are evaluated at specific points. Define A = f ^ (1) and B = f ^ (2) , thus f (x) = 3x^2 + 2A x + B . Differentiating yields f ^ (x) = 6x + 2A and further, f ^ (x) = 6 . From f ^ (x) = 6 , it follows that B = 6 . Substituting x=1 into f ^ (x) gives A = 6(1) + 2A , leading to A = -6 . The function becomes f (x) = 3x^2 - 12x + 6 , which corresponds to option C. C

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