Chapter 3.4, Problem 2QQ

To determine

To identify : the value of $g^{\prime }\left(2\right)$ .

1. $\frac{1}{13}$
2. $\frac{1}{4}$
3. $\frac{7}{4}$
4. $4$

Answer to Problem 2QQ

Correct option is (B).

Explanation of Solution

Given information :

A function $f\left(x\right)=x^3+x$ , $g\left(x\right)=f^{-1}\left(x\right)$ and, $g\left(2\right)=1$ .

Differentiate $g\left(x\right)=f^{-1}\left(x\right)$ with respect to $x$ .

  $\begin{array}{c}{g}^{\prime }\left(x\right)=\frac{d}{dx}\left[f^{-1}\left(x\right)\right]\\ =\frac{1}{f^{\prime }\left(f^{-1}\left(x\right)\right)}\\ =\frac{1}{f^{\prime }\left(g\left(x\right)\right)}\end{array}$

Substitute $2$ in place of $x$ in $g^{\prime }\left(x\right)=\frac{1}{f^{\prime }\left(g\left(x\right)\right)}$ .

  $\begin{array}{c}{g}^{\prime }\left(2\right)=\frac{1}{f^{\prime }\left(g\left(2\right)\right)}\\ =\frac{1}{f^{\prime }\left(1\right)}\end{array}$

Differentiate $f\left(x\right)=x^3+x$ with respect to $x$ .

  $\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}\left(x^3+x\right)\\ =3x^2+1\end{array}$

Substitute $1$ in place of $x$ in $f^{\prime }\left(x\right)=3x^2+1$ .

  $\begin{array}{c}{f}^{\prime }\left(1\right)=3{\left(1\right)}^2+1\\ =3+1\\ =4\end{array}$

Plug value of $f^{\prime }\left(1\right)=4$ in $g^{\prime }\left(2\right)=\frac{1}{f^{\prime }\left(1\right)}$ .

  $g^{\prime }\left(2\right)=\frac{1}{4}$

Correct option is (B).