If f''(x) =- f(x) and g(x) = f'(x) and `F(x)=(f((x)/(2)))^(2)+(g((x)/(2)))^(2)`
and given that F(5) =5, then F(10) is

Options

A. 15
B. 10
C. 0
D. 15

Answer

A

Explanation

We have
`f''(x)=-f(x) and g(x)=f'(x) Rightarrow g'(x)=-f(x) and g(x)=f'(x)`
Now,
`F(x)={{:(F((x)/(2))}^(2)+{g((x)/(2))}^(2)`
`Rightarrow F'(x)=f((x)/(2)) f'((x)/(2))+g((x)/(2))g'((x)/(2))`
`Rightarrow=F'(x)=f((x)/2) g((x)/(2))-g((x)/(2)) f((x)/(2))=0`
`Rightarrow F'(x)="Constant" Rightarrow F(10)=F(5)=5`