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If f(x) satisfies the relation 2 f(x)+f(1-x)=x² for all real x , then f(x) is

Options

  1. Ax²+2 x-1 6
  2. Bx²+2 x-1 3
  3. Cx²+4 x-1 3
  4. Dx²+4 x-1 6

Correct answer

B. x²+2 x-1 3

Step-by-step solution

Given the equation 2f(x) + f(1-x) = x^2 (1). Replacing x with 1-x in equation (1), we get: 2f(1-x) + f(x) = (1-x)^2 = 1 - 2x + x^2 (2). From (1), f(1-x) = x^2 - 2f(x) . Substituting this into (2): 2(x^2 - 2f(x)) + f(x) = 1 - 2x + x^2 2x^2 - 4f(x) + f(x) = 1 - 2x + x^2 -3f(x) = 1 - 2x + x^2 - 2x^2 -3f(x) = -x^2 - 2x + 1 3f(x) = x^2 + 2x - 1 f(x) = x^2 + 2x - 1 3 Answer: x²+2 x-1 3

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