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If f(x) and g(x) are two differentiable ...

If f(x) and g(x) are two differentiable functions on `R^+` such that xf'(x)+g(x)=0 and xg'(x)+f(x)=0 for all `x in R^+`and f(1)+g(1)=4, then the value of f"(2).g"(2) is

Text Solution

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`x*f'(x)+xg'(x)+g(x)|f(x)=0`
`d/dx[x(f(x)+g(x)]=0`
`f(x)+g(1)=4`
`a+b=4`
`b=4-x`
`xf'(x)+g(x)=0`
`x*-a/x^2+(9-a)/x=0`
`-a/x+(4-a)/x=0`
...
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