Question

Let $f: R \rightarrow R$ be defined by $f(x)=x^3+3 x+1$ and $g$ is the inverse of $f$ then the value of $g^{\prime \prime}(5)$ is equal to

• A. $\frac{-1}{36}$
• B. $\frac{-1}{6}$
• C. $\frac{1}{6}$
• D. $\frac{1}{36}$

Answer 1

Answer: (A)

$ g(5)=1 \text { and } f(1)=5 $
$\text { Now, } g^{\prime \prime}(5)=-\frac{f^{\prime \prime}(g(x))}{\left[f^{\prime}(g(x))\right]^3}$
$ \Rightarrow g^{\prime \prime}(5)=-\frac{f^{\prime \prime}(g(5))}{\left[f^{\prime}(g(5))\right]^3}=-\frac{f^{\prime \prime}(1)}{\left[f^{\prime}(1)\right]^3} $
$f^{\prime}(x)=3 x^2+3, f^{\prime}(1)=6 $
$f^{\prime \prime}(x)=6 x, f^{\prime \prime}(1)=6 $
$g^{\prime \prime}(5)=\frac{-6}{216}=\frac{-1}{36}$