Question

If $f^{\prime \prime }\left(x\right)=-f\left(x\right)$ 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 $5$

$f^{\prime \prime }\left(x\right)=-f\left(x\right)$ and $f^{\prime }\left(x\right)=g\left(x\right)$
$\Rightarrow f^{\prime \prime }\left(x\right).f^{\prime }\left(x\right)+f\left(x\right).f^{\prime }\left(x\right)=0$

$Integrating,$

$\left(f^{\prime }\right(x){)}^2+(f\left(x\right){)}^2=c\Rightarrow \left(f\right(x{)}^2+\left(g\right(x){)}^2=c$

$\Rightarrow F\left(x\right)=c$

$\Rightarrow F\left(10\right)=5$