Question

Let $f\left(x\right)=e^{(x^3+x^2+x)}$ for any real number x and let g be the inverse function for f. The value of $g^{\prime }\left(e^3\right)$ is

• A. $\frac{1}{6e^3}$
• B. $\frac{1}{6}$
• C. $\frac{1}{34e^{39}}$
• D. $6$

Answer 1

The correct option is A $\frac{1}{6e^3}$

$y=e^{x^3+x^2+x}$,

$\frac{dy}{dx}=e^{x^3+x^2+x}\cdot (3x^2+2x+1)$
$g^{\prime }\left(y\right)=\frac{dx}{dy}=\frac{1}{e^{x^3+x^2+x}\cdot (3x^2+2x+1)}$

$y=e^3\Rightarrow e^3=e^{x^3+x^2+x}$
$\Rightarrow x^3+x^2+x-3=0$
$\Rightarrow (x-1)(x^2+2x+3)=0$
$\Rightarrow x=1$
$\therefore g^{\prime }\left(e^3\right)=\frac{1}{6e^3}$
Hence, (A) is the correct answer.