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Question

Let f be a bijection satisfying f(x)=f(x). Then, (f1)′′(x) is equal to

A
1x3
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B
1x2
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C
f(x)
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D
f1(x)
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Solution

The correct option is C 1x2
We have,
f1(f(x))=x for all x
(f1)(f(x))f(x)=1 for all x
(f1)(f(x))=1f(x) for all x[f(x)=f(x)]
(f1)′′(f(x))f(x)=1|f(x)|2f(x)[ Differentiating both sides w.r.t. x]
(f1)′′(f(x))=1|f(x)|2
(f1)′′(x)=1x2

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