Question

If the function $f(x)=-4 e^{\frac{1-x}{2}}+1+x+\frac{x^2}{2}+\frac{x^3}{3}$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}\left(\frac{-7}{6}\right)$ equals

• A. $\frac{1}{5}$
• B. $-\frac{1}{5}$
• C. $\frac{6}{7}$
• D. $-\frac{6}{7}$

Answer 1

Answer: (A)

$\frac{ dy }{ dx }=\frac{4}{2} e ^{\frac{1- x }{2}}+1+ x + x ^2=2 e ^{\left(\frac{1- x }{2}\right)}+ x ^2+ x +1$
$g ^{\prime}( y )=\frac{ dx }{ dy }=\frac{1}{2 e ^{-\left(\frac{ x -1}{2}\right)}+ x ^2+ x +1}$, when $y =-\frac{7}{6}$ then $x =1$
$\left.\frac{ dy }{ dx }\right]_{ x =-7 / 6}=\frac{1}{2+3}=\frac{1}{5}$