Question

Let $f:R\to R$ be defined by $f(x)=x^3+3x+1$ 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)

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