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Suppose a function f satisfies f'(x) = -f(x). Define g(x) = f^x, and let F(x) = \left(\int f\left(\frac{x}{2}\right)\right)^2 + \left(\int g\left(\frac{x}{2}\right)\right)^2. If F(5) = 5, what is the value of F(10)?

1. 5
2. 10
3. 0
4. 15

Correct answer: 10

Solution

The function F(x) is defined in terms of integrals of f and g, both of which are related through their definitions and the properties of their derivatives. Since F(5) = 5 and the integrals are evaluated at half the input, it follows that F(10) will be double F(5), resulting in F(10) = 10.

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