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If f''(x) =- f(x) and g(x) = f'(x) and F...

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

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Verified by Experts

The correct Answer is:
5
`F^(')(x)=[f(x/2).f^(')(x/2)+g(x/2).g^(')(x/2)]`
Here, `g(x)=f^(')(x)` and `g^(')(x)=-f(x)=f^(")(x)`
So, `F^(')(x)=f(x/2).g(x/2)-f(x/2).g(x/2)=0`
Hence `F^'(x)` is a constant function therefore `f(10)=5`
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