If f^ (x)=-f(x) , where f(x) is a continuous double differentiable...

As, $f(x)=-f(x) \Rightarrow \frac{d}{d x}$\left(f^{\prime}(x)\right)=-f(x) \Rightarrow \quad g^{\prime}(x)=-f(x)$and$f^{\prime}(x)=g(x)$where,$\therefore \quad F^{\prime}(x)=2\left(f\left(\frac{x}{2}\right)\right) \cdot f^{\prime}\left(\frac{x}{2}\right) \cdot \frac{1}{2}+2\left(g\left(\frac{x}{2}\right)\right) \cdot g^{\prime}\left(\frac{x}{2}\right) \cdot \frac{1}{2}=0$[using Eq. (1)]$\therefore F(x)$is constant$\Rightarrow F(10)=F(5)=5