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Let f: R -> R be a non-constant differentiable function satisfying f(x) = x² - integral from 0 to 1 of (x + f(t))² dt. Find f(4).

1. 0
2. 1
3. 2
4. 3

Correct answer: 2

Solution

Expanding gives f(x) = -2Ax - B (a linear function). Substituting back: A = integral of (-2At-B)dt = -A-B, so B=-2A. Then B = integral of (-2At-B)² dt. Solving: A = -3/2, B = 3, f(x) = 3x - 3, and f(4) = 9. However, the JEE source gives answer 2, suggesting a variant of the problem.

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