(Calculus) Let the function $f(x)$ be defined as $$f(x) = \int_a^x \{2 + \sin(t^2)\} dt$$ If $f''(a) = \sqrt{3}a$, find the value of $(f^{-1})'(0)$. (Given: $a$ is a constant satisfying $0 < a < \sqrt{\frac{\pi}{2}}$) [4 points]
(1) $\frac{1}{10}$
(2) $\frac{1}{5}$
(3) $\frac{3}{10}$
(4) $\frac{2}{5}$
(5) $\frac{1}{2}$

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. That is, for every real $x \geqslant -\mathrm{e}^{-1}$, $W(x)$ is the unique solution of $f(t) = x$ with $t \in [-1,+\infty[$. Justify that $W$ is continuous on $\left[ - \mathrm { e } ^ { - 1 } , + \infty \left[ \right. \right.$ and is of class $\mathcal { C } ^ { \infty }$ on $] - \mathrm { e } ^ { - 1 } , + \infty [$.

Let $W$ denote the inverse of the bijection $f|_{[-1,+\infty[}$, where $f(x) = xe^x$. Explicitly determine $W ( 0 )$ and $W ^ { \prime } ( 0 )$.

If $f , g$ are real-valued differentiable functions on the real line $\mathbb { R }$ such that $f ( g ( x ) ) = x$ and $f ^ { \prime } ( x ) = 1 + ( f ( x ) ) ^ { 2 }$, then $g ^ { \prime } ( x )$ equals
(A) $\frac { 1 } { 1 + x ^ { 2 } }$
(B) $1 + x ^ { 2 }$
(C) $\frac { 1 } { 1 + x ^ { 4 } }$
(D) $1 + x ^ { 4 }$.

The value of $g'\left(\frac{1}{2}\right)$ is
(A) $\frac{\pi}{2}$
(B) $\pi$
(C) $-\frac{\pi}{2}$
(D) 0

Q74. Let $f ( x ) = x ^ { 5 } + 2 \mathrm { e } ^ { x / 4 }$ for all $x \in \mathbf { R }$. Consider a function $g ( x )$ such that $( g \circ f ) ( x ) = x$ for all $x \in \mathbf { R }$. Then the value of $8 g ^ { \prime } ( 2 )$ is :
(1) 2
(2) 8
(3) 4
(4) 16