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) $\psi _ { 1 } ( x ) \leq 1$, for all $x > 0$\\
(B) $\psi _ { 2 } ( x ) \leq 0$, for all $x > 0$\\
(C) $f ( x ) \geq 1 - e ^ { - x ^ { 2 } } - \frac { 2 } { 3 } x ^ { 3 } + \frac { 2 } { 5 } x ^ { 5 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$\\
(D) $g ( x ) \leq \frac { 2 } { 3 } x ^ { 3 } - \frac { 2 } { 5 } x ^ { 5 } + \frac { 1 } { 7 } x ^ { 7 }$, for all $x \in \left( 0 , \frac { 1 } { 2 } \right)$