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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
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