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Let `f` be a real-valued function defined on the inverval `(-1,1)` such that `e^(-x)f(x)=2+int_0^xsqrt(t^4+1)dt ,` for all, `x in (-1,1)a n dl e tf^(-1)` be the inverse function of `fdot` Then `(f^(-1))^'(2)` is equal to 1 (b) `1/3` (c) `1/2` (d) `1/e`

A

`1`

B

`1//3`

C

`1//2`

D

`1//e`

Text Solution

Verified by Experts

The correct Answer is:
B
`e^(-x)f(x)=2+int_(0)^(x)sqrt(t^(4)+1)dt`…………..1
Now `f(f^(-1)(x))=x`
`:.f'(f^(-1)(x))(f^(-1)(x))'=1`
or `(f^(-1))'(2)=1/(f'(f^(-1)(2)))`
From 1
`f(0)=2` or `f^(-1)(2)=0`
or `(f^(-1))'(2)=1/(f'(0))`
`e^(-x)f(x)=2+int_(0)^(x)sqrt(t^(4)+1)dt`
Differentiating w.r.t `x`
or `e^(-x)(f'(x)-f(x))=sqrt(x^(4)+1)`
put `x=0`
`:.f'(0)-2=1`
or `f'(0)=3`
`(f^(-1))'(2)=1//3`
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