Question

If $f\left(x\right)=x^3+e^{x/2}$ then $g"\left(1\right)$ is equal to, (where $f\left(x\right)$ and $g\left(x\right)$ are inverse functions of each other).

• A. $2$
• B. $-2$
• C. $-\frac{1}{2}$
• D. $\frac{1}{2}$

Answer 1

Answer: (B)

$-2$

$g\left(f\right(x\left)\right)=x$

$g^{\prime }f\left(x\right)=1/f^{\prime }\left(X\right)$

$g^{\prime \prime }f\left(x\right)=\frac{f^{\prime \prime }\left(x\right)}{{f^{\prime }\left(x\right)}^3}$

$f\left(x\right)=1atx=0$

$f\left(0\right)=1$

$f^{\prime }\left(x\right)=3x^2+e^{\left(x/2\right)}/2$

$f^{\prime }\left(0\right)=1/2$

$f^{\prime \prime }\left(x\right)=6x+e^{x/2}/4$

$f^{\prime \prime }\left(0\right)=1/4$

$g^{\prime \prime }\left(1\right)=\frac{-1/4}{1/8}=-2$