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If f(x)=int(0)^(x) {f(t)}^(-1)dt, " and ...

If f(x)`=int_(0)^(x) {f(t)}^(-1)dt, " and " int_(0)^(1) {f(t}^(-1)dt=sqrt(2)`, then f(x)=

A

`sqrt(2x)`

B

`sqrt(2 log_(e )x)`

C

`sqrt(3x-1)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
f(f)`=overset(x)underset(0)int {f(t)}^(-1)dt`
`rArr f'(x)={f(t)}^(-1)`
`rArr f'(x)f(x)=1`
`rArr 2f(x)f'(x)=2 rArr{f(x)}^(2)=2x+C rArrf(x)=sqrt(2x+C)`b
Now,
`f(1)=sqrt(2)rArr sqrt(2)=sqrt(2+C) rArr C=0`
`:. F(x)=sqrt(x)`
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