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Let f(x) be differentiable on the interv...

Let `f(x)` be differentiable on the interval `(0,oo)` such that `f(1)=1` and `lim_(t->x) (t^2f(x)-x^2f(t))/(t-x)=1` for each `x>0`. Then `f(x)=`

A

(a) `1/(3x)=(2x^(2))/3`

B

(b) `-1/(3x)+(4x^(2))/3`

C

(c) `-1/x+2/x^(2)`

D

(d) 1/x

Text Solution

Verified by Experts

Given `lim_(t rarr x) (t^(2)f(x)-x^(2)f(t))/(t-x)=1`
` rArr x^(2)f'(x)-2xf(x)+1=0`
` rArr (x^(2)f'(x)-2xf(x))/((x^(2))^(2))+1/x^(4)=0`
` rArr d/dx((f(x))/(x^(2)))=-1/x^(4)`
On integrating both sides, we get
`f(x)=cx^(2)+1/(3x)`
Also,`f(1)=1, c=2/3`
Hence, `f(x)=2/3x^(2)+1/(3x)`
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