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A function f: R -> (0, infinity) satisfies f(x + y) = f(x) * f(y) for all real x, y. Given that f(x) is non-zero everywhere, differentiable on R, and f'(0) = 2, find f(x).

1. e^(-x)
2. e^(2x)
3. e^(x)
4. e^(-2x)

Correct answer: e^(2x)

Solution

The only continuous non-zero solution to f(x+y)=f(x)*f(y) is f(x)=e^(kx). Using f'(0)=k=2 gives f(x)=e^(2x).

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