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kanika kaul Grade: 12
        

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).....

6 years ago

Answers : (1)

Puneet Mittal AskiitiansExpert-IITD
22 Points
										

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

6 years ago
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