Question

If the function $f(x)=x^3+e^2$ and $g(x)=f^{-1}(x)$, then the value of $g^{\prime}(1)$ is $(4,-1), 80]$

Answer 1

SOLUTION : $g(f(x))=x \quad \Rightarrow \quad g^{\prime}(f(x)) f^{\prime}(x)=1$

if $f(x)=1 \quad \Rightarrow \quad x=0, f(0)=1$

substitute $x=0$ in (i), we get

$g^{\prime}(1)=\frac{1}{f^{\prime}(0)} \Rightarrow g^{\prime}(1)=2 \quad \Rightarrow \quad\left(f^{\prime}(x)=3 x^2+\frac{1}{2} e^{x / 2} \Rightarrow f^{\prime}(0)=\frac{1}{2}\right)$