Mathematics - Differentiation Question with Solution | TestHub

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