Question

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

• A. 5
• B. 10
• C. 15

Answer 1

The correct option is A 5

$f^{\prime }\left(x\right)=-f\left(x\right)and{f}^{\prime }\left(x\right)=g\left(x\right)$
$\Rightarrow f^{\prime \prime }\left(x\right).f^{\prime }\left(x\right)+f\left(x\right).f^{\prime }\left(x\right)=0$
$\Rightarrow f(x{)}^2+(f^{\prime }\left(x\right){)}^2=c\Rightarrow \left(f\right(x{)}^2+\left(g\right(x\left)\right)2=c$
$\Rightarrow F\left(x\right)=c\Rightarrow F\left(10\right)=5$
Hence (A) is the correct answer.