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A differentiable function f(x) satisfies 6x * integral from 0 to 1 of f(tx) dt = 2x³ - 3x² + 6x + 5 for all real x. Which of the following statements about f(x) is/are correct?

1. f(x) is symmetric about x = 1/2
2. f(x) = 0 has no real roots
3. f(-x) = f(x + 1) for all real x
4. f(x) = 1/2 has two real and equal roots

Correct answer: f(-x) = f(x + 1) for all real x

Solution

Substituting u = tx reduces the integral equation to 6*F(x) = 2x³ - 3x² + 6x + 5, where F is the antiderivative of f. Differentiating gives f(x) = x² - x + 1. Since f(-x) = x² + x + 1 = f(x+1), statement C holds. The discriminant of f(x)=0 is negative (no real roots, statement B also holds), and f(x) is symmetric about x = 1/2 (not x = 1, so A is wrong). f(x) = 1/2 has no real roots (discriminant = 1 - 4*(1/2) = -1 < 0), so D is wrong.

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