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Let f: [0, infinity) -> R be continuous and satisfy f(x) = 1 - 2x + integral₀^x e^(x-t) f(t) dt for all x >= 0. Which of the following statement(s) is/are TRUE?

1. The curve y = f(x) passes through (1, 2)
2. The curve y = f(x) passes through (2, -1)
3. The area of the region {(x, y) in [0,1] x R: f(x) <= y <= sqrt(1 - x²)} is (pi - 2)/4
4. The area of the region {(x, y) in [0,1] x R: f(x) <= y <= sqrt(1 - x²)} is (pi - 1)/4

Correct answer: The curve y = f(x) passes through (2, -1)

Solution

Differentiating gives f' - 2f = 2x - 3 with f(0) = 1, whose solution is f(x) = 1 - x; this passes through (2, -1) and makes the enclosed area (pi - 2)/4, so the (2,-1) and (pi-2)/4 statements are true.

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