JEE MAIN - Mathematics (2022 - 25th June Morning Shift - No. 4)
Let f : R $$\to$$ R be defined as $$f(x) = {x^3} + x - 5$$. If g(x) is a function such that $$f(g(x)) = x,\forall 'x' \in R$$, then g'(63) is equal to ________________.
$${1 \over {49}}$$
$${3 \over {49}}$$
$${43 \over {49}}$$
$${91 \over {49}}$$
Explanation
$$f(x) = 3{x^2} + 1$$
f'(x) is bijective function
and $$f(g(x)) = x \Rightarrow g(x)$$ is inverse of f(x)
$$g(f(x)) = x$$
$$g'(f(x))\,.\,f'(x) = 1$$
$$g'(f(x)) = {1 \over {3{x^2} + 1}}$$
Put x = 4 we get
$$g'(63) = {1 \over {49}}$$
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