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A continuous function f(x) takes positive values for x >= 0 and satisfies the integral equation integral[0 to x] f(t) dt = x * sqrt(f(x)), with f(1) = 1/2. Find f(sqrt(2) + 1).

1. 1
2. sqrt(2) - 1
3. 1/4
4. 1/(sqrt(2) - 1)

Correct answer: 1/4

Solution

Differentiating both sides and substituting h = sqrt(f) yields the separable ODE h(h-1) = xh'. Integrating gives (h-1)/h = Ax. Using f(1) = 1/2 (so h(1) = 1/sqrt(2)) gives A = 1 - sqrt(2). At x = sqrt(2)+1, this gives h = 1/2, so f = h² = 1/4.

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