Question

Let $f : R \rightarrow R$ 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 $g ^{\prime}(63)$ is equal to ______.

• A. $\frac{1}{49}$
• B. $\frac{3}{49}$
• C. $\frac{43}{49}$
• D. $\frac{91}{49}$

Answer 1

Answer: (A)

$f(x)=x^{3}+x-5$
$\Rightarrow f^{\prime}(x)=3 x^{2}+1 \Rightarrow$ increasing function
$\Rightarrow $ invertible
$\Rightarrow g(x)$ is inverse of $f(x)$
$\Rightarrow g(f(x))=x$
$\Rightarrow g^{\prime}(f(x)) f^{\prime}(x)=1$
$\Rightarrow f(x)=63$
$\Rightarrow x^{3}+x-5=63$
$\Rightarrow x=4$
put $x=4$
$g^{\prime}(f(4)) f^{\prime}(4)=1$
$g^{\prime}(63) \times 49=1 \left\{f^{\prime}(4)=49\right\}$
$ g^{\prime}(63)=\frac{1}{49}$