If ` f: (0, oo) ->(0, oo)` for which there is a positive real number a such that it satisfies differential equation `f'(a/x)=x/(f(x))` , then

Options

A. `f(x)` can be linear
B. `f(x)` can be a functional of the type `m(x)^(1//n),m epsilonR^(+),n epsilonI^(+)`
C. `f^(')(x)` can be positive
D. `f(x)` can be twice differentiable

Answer

A, B, C, D

Explanation

`g(x)=f(x).f(a/x)`
`g^(')(x)=f^(')(x).f(a/x)-f(x)f^(')(a/x)(a/(x^(2)))=a/x-x(a/(x^(2)))=0`
`impliesg(x)=b`
`implies f(x).f(a/x)=b`
`impliesf(x)=sqrt(b)(x/(sqrt(a)))^(a/b)`