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If f(x)+2 f(1-x)=x^2+5 real values of x , then f(x) is given by

Options

  1. Ax^2-5
  2. B2
  3. C(x-2)^2+3 3
  4. DNone of these

Correct answer

C. (x-2)^2+3 3

Step-by-step solution

Given the functional equation f(x) + 2f(1-x) = x^2 + 5 . Replace x with 1-x in the equation: f(1-x) + 2f(1-(1-x)) = (1-x)^2 + 5 f(1-x) + 2f(x) = (1-x)^2 + 5 We now have a system of two linear equations in f(x) and f(1-x) : 1) f(x) + 2f(1-x) = x^2 + 5 2) 2f(x) + f(1-x) = (1-x)^2 + 5 = x^2 - 2x + 6 Multiply equation (2) by 2: 4f(x) + 2f(1-x) = 2x^2 - 4x + 12 Subtract equation (1) from this result: (4f(x) + 2f(1-x)) - (f(x) + 2f(1-x)) = (2x^2 - 4x + 12) - (x^2 + 5) 3f(x) = x^2 - 4x + 7 f(x) = x^2 - 4x + 7 3 Rewrite th

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