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A curve y = f(x) passes through (0, 1) and the curve y = integral from -infinity to x of f(t) dt passes through (0, 1/2). At points with equal x-coordinate, the tangents to the two curves meet on the x-axis. Find f(x).

1. log(x + 2)
2. e^(2x)
3. (3x² - x³) / x^(3/2)
4. log((x² - x³) / x^(1/2))

Correct answer: e^(2x)

Solution

Equal x-intercepts force f'/f = g'/g; since g' = f this yields g = A e^(kx), f = k g, and the two points fix k = 2, giving f(x) = e^(2x).

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