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Suppose f satisfies x⁴*f(x) - sqrt(1 - sin(2*pi*x)) = |f(x)| - 2*f(x) for all x. Find f(-2).

1. 1/17
2. 1/11
3. 1/19
4. 0

Correct answer: 1/17

Solution

At x = -2: x⁴ = 16, and sin(2*pi*(-2)) = sin(-4*pi) = 0, so sqrt(1 - 0) = 1. The equation becomes 16*f(-2) - 1 = |f(-2)| - 2*f(-2). Assuming f(-2) >= 0 gives |f(-2)| = f(-2), so 16f - 1 = f - 2f = -f, hence 17f = 1 and f = 1/17, which is nonnegative and therefore consistent. So f(-2) = 1/17.

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