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Let f:(0,oo)->R be a differentiable func...

Let `f:(0,oo)->R` be a differentiable function such that `f'(x)=2-f(x)/x` for all `x in (0,oo)` and `f(1)=1`, then

A

`underset(xrarr0^(+))limf'((1)/(x))=1`

B

`underset(xrarr0^(+))limxf((1)/(x))=2`

C

`underset(xrarr0^(+))limx^(2)f'(x)=0`

D

`|f(x)|le2" for all "x in (0,2)`

Text Solution

Verified by Experts

`f'(x)+(f(x))/(x)=2`
`rArr" "xf'(x)+f(x)=2x`
`rArr" "int d(x.f(x))=int 2xdx`
`rArr" "xf(x)=x+(c)/(x)" "(c ne 0 as f(1) ne1)`
`underset(xrarr0^(+))limf'((1)/(x))=underset(xrarr0^(+))lim(1-cx^(2))=1`
`underset(xrarr0^(+))limxf((1)/(x))=underset(xrarr0^(+))lim(1+cx^(2))=1`
`underset(xrarr0^(+))limx^(2)f'(x)=underset(xrarr0^(+))lim(x^(2)-c)=-c ne 0`
`underset(xrarr0^(+))limf(x)=oo or -oo" so option (4) is incorrect "`
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