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Let f :(0, infty)to R be a differentiabl...

Let `f :(0, infty)to R` be a differentiable function such that `f'(x)=2-(f(x))/x` for all `x in (0,infty)` and `f (1)ne1`. Then (a) `lim_(x to 0+)f'(1/x)=1` (b) `lim_(x to 0+)x f'(1/x)=2` (c) `lim_(x to 0+)x^2f'(x)=0` (d) `abs(f(x))le 2` for all `x in(0,2)`

A

(a) `lim_(x to 0+)f'(1/x)=1`

B

(b) `lim_(x to 0+)x f'(1/x)=2`

C

(c) `lim_(x to 0+)x^2f'(x)=0`

D

(d) `abs(f(x))le 2` for all `x in(0,2)`

Text Solution

Verified by Experts

Here, `f'(x)=2-(f(x))/x`
or `dy/dx+y/x=2` [i.e. linear differential equation in y ]
Integrating Factor, `IF= e^(int1/xdx)=e^(logx)=x`
`therefore` Required solutionn is `y cdot (IF) = int Q(IF)dx=C`
`rArr y(x)=int 2(x)dx+C`
`rArr yx=x^(2)+c`
`therefore y=x+C/x [therefore C ne 0,as f(1)ne1]`
(a) `lim_(xrarr0^(+))f'(1/x)=lim_(xrarr0^(+)) (1-Cx^(2))=1`
`therefore` Option (a) is correct.
(b) `lim_(xrarr0^(+))x f(1/x)=lim_(xrarr0^(+)) (1-Cx^(2))=1`
`therefore` Option (b) is incorrect.
`lim_(xrarr0^(+))x^(2) f'(x)=lim_(xrarr0^(+)) (x^(2)-C)=-C ne 0`
`therefore` (c) is incorrect.
(d) `f(x)=x+C/x,C ne 0`
For `C gt0, lim_(x rarr 0^(+)) f(x)=infty `
`therefore` Function is not bounded in (0,2).
`therefore` Option (d) is incorrect.
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