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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

From $f'(x)=-f(x)$, the function is of exponential type, so the integrals in $F(x)$ scale predictably with the argument. The given condition $F(5)=5$ implies the same linear scaling, hence doubling the input doubles the value: $F(10)=10$.

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