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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

`(3)/(1+9x^(3))`

B

`(9)/(1+9x^(3))`

C

`(3xsqrt(x))/(1-9x^(3))`

D

`(3x)/(1-9x^(3))`

Text Solution

Verified by Experts

We have
`y=tan^(-1)((6xsqrt(x))/(1-9x^(3)))`
`=tan^(-1)((2cdot(3x^(3//2)))/(1-(3x^(3//2))^(2)))=2 tan ^(-1)(3x^(3//2))`
`therefore" "(dy)/(dx)=2xx(1)/(1+9x^(3))xx3xx(3)/(2)xxx^(1//2)=(9sqrt(x))/(1+9x^(3))`
`therefore" "g(x)=(9)/(1+9x^(3))`
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