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Let `f:square to square, g : square to square` and `h: square to square` be differentiable functions such that `f(x)=x^(3)+3x=2, g(f(x))=x` and `h(g(g(x)))=x` for all `x epsilonR`. Then

A

666

B

16

C

66

D

111

Text Solution

Verified by Experts

The correct Answer is:
A
We have
`g(f(x))=x and h(g(g(x))=x"for all "x in R`
`Rightarrow h(g(g(f(x)))=f(x)"for all x"in R" "["Replacing x by f(x)"]`
`Rightarrow h(g(x))=f(x)"for all x"in R " "[therefore g(f(x))=x]`
`Rightarrow h(g(f(x)))=f(f(x))"for all "x in R " "["Replacing x by f(x))"]`
`Rightarrow h(x)=f(f(x))"for all x"in R" "[therefore g(f(x))=x]`
`Rightarrow h'(x)=f'(f(x)) f'(x)"for all "x in R`
`Rightarrow h'(1)=f'(f(1)) f'(1)=f'(6)f'(1) " "[therefore f'(x)=3x^(2)+3]`
`h'(1)=(3xx26+3) (3+3)=666`
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