IF f= -f(x) where f(x) is a continuous double differentiable function and g(x)=f'(x)

If F(x) =(f(x/2))square+(g(x/2))square and F(5) =5 then find F(10).....

Hi Kanika,

You can calculate answer as follows :

F(x) = f(x/2)²  + g(x/2)²

Differentiating w.r.t x,

F'(x) = 2*f(x/2)*f'(x/2)*1/2   +  2*g(x/2)*g'(x/2)*1/2 

       = f(x/2)*f'(x/2)  + g(x/2)*g'(x/2)

Now g(x) = f'(x)

Differentiating , g'(x) = f''(x)

                               = - f(x) ( as given )

Thus F'(x) =  f(x/2)*f'(x/2) + f'(x/2)*( - f(x/2) )

               = f(x/2)*f'(x/2)  - f'(x/2)*f(x/2)

               = 0

This Implies F(x) is a constant

Let F(x) = C

Now F(5) = 5 ( given )

This Implied C = 5.

Hence F(x) = 5.

So F(10) = 5 is the Answer

Please feel free to ask as many question you have.

Puneet