Question

Let $f : R \rightarrow R$ be a function defined as $f ( x )= e ^{ x ^2}\left( e ^{ x ^3+ x +1}- e ^{- x ^3- x -3}\right)+2 x +5$ and $g$ is the inverse function of $f$, then

• A. $\frac{d}{d x}(x f(g(x)))|_{x=5}=10$
• B. $\frac{d}{d x}(g(x))|_{x=3}=\frac{1}{10}$
• C. $\left.\frac{ d }{ dx }\left(\frac{ x }{ g ( x )}\right)\right|_{ x =3}=-31$
• D. $\left.\frac{ d }{ dx }\left(\frac{ g ( f ( x )))}{ x ^2}\right)\right|_{ x = e }=1$

Answer 1

Answer: (A), (B)

$ f ( x )= e ^{ x ^3+ x ^2+ x +1}- e ^{- x ^3+ x ^2- x -3}+2 x +5$
$f (-1)=3$
$f ^{\prime}( x )= e ^{ x ^3+ x ^2+ x +1}\left(3 x ^2+2 x +1\right)- e ^{- x ^3+ x ^2- x -3}\left(-3 x ^2+2 x -1\right)+2 $
$f ^{\prime}(-1)=3-2+1-(-3-2-1)+2=10 $
$\text { Now, verify the options. }$