If the functions `f(x)=x^(3)+e^(x//2) " and " g(x)=f^(-1)(x)`, the value of g'(1) is ………… .

Options

A. `(1)/(2)`
B. 2
C. 1
D. `-(1)/(2)`

Answer

B

Explanation

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`