Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$, $$\begin{gathered}
\psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\
\psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\
f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0
\end{gathered}$$ and $$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$ Which of the following statements is TRUE ? (A) $f ( \sqrt { \ln 3 } ) + g ( \sqrt { \ln 3 } ) = \frac { 1 } { 3 }$ (B) For every $x > 1$, there exists an $\alpha \in ( 1 , x )$ such that $\psi _ { 1 } ( x ) = 1 + \alpha x$ (C) For every $x > 0$, there exists a $\beta \in ( 0 , x )$ such that $\psi _ { 2 } ( x ) = 2 x \left( \psi _ { 1 } ( \beta ) - 1 \right)$ (D) $f$ is an increasing function on the interval $\left[ 0 , \frac { 3 } { 2 } \right]$
Let $\psi _ { 1 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad \psi _ { 2 } : [ 0 , \infty ) \rightarrow \mathbb { R } , \quad f : [ 0 , \infty ) \rightarrow \mathbb { R }$ and $g : [ 0 , \infty ) \rightarrow \mathbb { R }$ be functions such that $f ( 0 ) = g ( 0 ) = 0$,
$$\begin{gathered}
\psi _ { 1 } ( x ) = e ^ { - x } + x , \quad x \geq 0 \\
\psi _ { 2 } ( x ) = x ^ { 2 } - 2 x - 2 e ^ { - x } + 2 , \quad x \geq 0 \\
f ( x ) = \int _ { - x } ^ { x } \left( | t | - t ^ { 2 } \right) e ^ { - t ^ { 2 } } d t , \quad x > 0
\end{gathered}$$
and
$$g ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt { t } e ^ { - t } d t , \quad x > 0$$
Which of the following statements is TRUE ?\\
(A) $f ( \sqrt { \ln 3 } ) + g ( \sqrt { \ln 3 } ) = \frac { 1 } { 3 }$\\
(B) For every $x > 1$, there exists an $\alpha \in ( 1 , x )$ such that $\psi _ { 1 } ( x ) = 1 + \alpha x$\\
(C) For every $x > 0$, there exists a $\beta \in ( 0 , x )$ such that $\psi _ { 2 } ( x ) = 2 x \left( \psi _ { 1 } ( \beta ) - 1 \right)$\\
(D) $f$ is an increasing function on the interval $\left[ 0 , \frac { 3 } { 2 } \right]$