Obviously, ${\log _{10}}\alpha ,\,{\log _3}\alpha ,\,{\log _e}\alpha ,\,{\log _2}\alpha $ are in increasing order.
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$f(0)=g(0)=0$
$\Psi_1( x )= e ^{- x }+ x , \quad x \geq 0$
$\Psi_2( x )= x ^2-2 x -2 e ^{- x }+2, x \geq 0$
$f( x )=\int_{- x }^{ x }\left(| t |- t ^2\right) e ^{- t ^2} dt , x >0$
and
$g(x)=\int_0^{x^2} \sqrt{t} e^{-t} d t, x>0$
($1$) 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 $\left[0, \frac{3}{2}\right]$
($2$) 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 $x \in\left(0, \frac{1}{2}\right)$