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Let f be a continuous function satisfyin...

Let f be a continuous function satisfying `f '(l n x)=[1` for `0< x<= 1, x` for `x > 1` and `f (0) = 0` then `f(x)` can be defined as

A

`f(x)={{:(1,if,xle1),(1-e^(x),if,xgt1):}`

B

`f(x)={{:(1,if,xle1),(e^(x)-1,if,xgt1):}`

C

`f(x)={{:(1,if,xlt1),(e^(x),if,xgt1):}`

D

`f(x)={{:(x,if,xle1),(e^(x)-1,if,xgt1):}`

Text Solution

Verified by Experts

The correct Answer is:
D
`f'(lnx)={{:(1,"for", 0ltxle1),(x,"for",xgt1):}`
Put log x = t
`rArr" "x=e^(t)`
For `x gt 1,f'(t)=e^(t),t gt0`
integrating `f(t)=e^(t)+C,`
`f(0)=e^(0)+c`
`rArr" "c=-1" (given f(0) = 0)"`
`therefore" "f(t)=e^(t)-1" for "tgt0" (corresponding to x gt 1)"`
Hence `f(x)=e^(x)-1" for "x gt 0" (1)"`
again for `0lt x le 1`
`f'(logx) = 1" "(x=e^(t))`
`f'(t)=1" for "t le0`
`f(t)=t+C`
`f(0)=0+C`
`rArr" C=0`
`rArr" "f(t)=t " for "t lt0`
`rArr" "f(x)=x" for "x le0`
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