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`

Options

A. 1
B. `1/3`
C. `1/2`
D. `1/e`

Answer

B

Explanation

Given `e^(-x)f(x)=2+int_(0)^(x)sqrt(t^(4)+1)dt, x epsilon(-1,1)`
On differentiating both sides w.r.t `x` we get
`e^(-x)(f'(x)-f(x))=sqrt(x^(4)+1)`
`impliesf'(x)=f(x)+sqrt(x^(4)+1)e^(x)`
Simcey `f^(-1)` is the inverse of `f`
`:.f^(-1)(f(x))=ximpliesf^(-1')(f(x))f'(x)=1`
`impliesf^(-1')(f(x))=1/(f'(x))`
`=f^(-1')(f(x))=1/(f(x)+sqrt(x^(4)+1)e^(x))`
At `x=0, f(x)=2`
`impliesf^(-1')(2)=1/(2+1)=1/3`