JEE MAIN - Mathematics (2020 - 9th January Evening Slot - No. 19)
Let ƒ and g be differentiable functions on R
such that fog is the identity function. If for some
a, b $$ \in $$ R, g'(a) = 5 and g(a) = b, then ƒ'(b) is
equal to :
1
5
$${2 \over 5}$$
$${1 \over 5}$$
Explanation
Given the function composition f(g(x)) is the identity function, it means f(g(x)) = x for all x.
$$ \Rightarrow $$ ƒ'(g(x)) g'(x) = 1
put x = a
$$ \Rightarrow $$ ƒ'(b) g'(a) = 1
$$ \Rightarrow $$ ƒ'(b) = $${1 \over 5}$$
$$ \Rightarrow $$ ƒ'(g(x)) g'(x) = 1
put x = a
$$ \Rightarrow $$ ƒ'(b) g'(a) = 1
$$ \Rightarrow $$ ƒ'(b) = $${1 \over 5}$$
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