Question

If $f"\left(x\right)=-f\left(x\right)$ and $g\left(x\right)=f^{\prime }\left(x\right)$ and $F\left(x\right)={\left(f\left(\frac{x}{2}\right)\right)}^2+{\left(g\left(\frac{x}{2}\right)\right)}^2$ and given that $F\left(5\right)=5$, then $F\left(10\right)$ is equal to

• A. $5$
• B. $10$
• C. $0$
• D. $15$

Answer 1

The correct option is A $5$

$as{f}^{\prime \prime }\left(x\right)=-f\left(x\right)\therefore f\left(x\right)=Ksinxandg\left(x\right)=f^{\prime }\left(x\right)=+KcosxnowF\left(5\right)=5{\left[Kisn\right(\frac{5}{2}\left)\right]}^2+{\left[Kcos\right(\frac{5}{2}\left)\right]}^2=5K^2=5thenF\left(10\right)=(Ksin5{)}^2+(Kcos5{)}^2=K^2=5$