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If the function f(x)=x^3+e^(x/2) and g(x...

If the function `f(x)=x^3+e^(x/2) and g(x)=f^(-1)(x)`, then the value of g'(1) is

A

`(1)/(2)`

B

2

C

1

D

`-(1)/(2)`

Text Solution

Verified by Experts

The correct Answer is:
B
It is given that g(x) is inverse of f(x).
`:." "fog(x)=x" for all "x`
`implies" "f'(g(x))g'(x)=1" for all "x`
`implies" "f'(g(1))g'(1)=1`
`implies" "g'(1)=(1)/(f'(g(1)))" "…(i)`
Again, g is inverse of f.
`:." "f(x)=yimpliesg(y)=x`
We have, `f(0)=1`
`:." "g(1)=0`
From (i), we get
`g'(1)=(1)/(f'(0))`
Now,
`f(x)=x^(3)+e^(x//2)impliesf'(x)=3x^(2)+(1)/(2)e^(x//2)impliesf'(0)=(1)/(2)`
Hence, `g'(1)=(1)/(1//2)=2`
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