Question

Let $\Psi_{1}:[0, \infty) \rightarrow R, \Psi_{2}:[0, \infty) \rightarrow R, f:[0, \infty) \rightarrow R$ and $g:[0, \infty) \rightarrow R$ be functions such that $f (0)= g (0)=0$,
$\psi_{1}(x)=e^{-x}+x, x \geq 0, $
$\psi_{2}(x)=x^{2}-2 x-2 e^{-x}+2, x \geq 0,$
$f(x)=\int\limits_{-x}^{x}\left(|t|-t^{2}\right) e^{-t^{2}} d t, x > 0$
and $g(x)=\int\limits_{0}^{x^{2}} \sqrt{t} e^{-t} d t, x>0$.
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 $x \in\left(0, \frac{1}{2}\right)$

Answer 1

Answer: (D)

$\psi^{\prime}{ }_{1}( x )=1- e ^{- x }>0$
$\psi_{2}( x )=( x -1)^{2}+1-2 e ^{- x }>0$ for $x =1$
$\because e ^{- t }=1- t +\frac{ t ^{2}}{2}-\frac{ t ^{3}}{3}+\ldots$
$ \sqrt{ t } e ^{- t }=\sqrt{ t }- t ^{3 / 2}+\frac{1}{2} t ^{5 / 2} \ldots$
So, $ \sqrt{ t } e ^{- t }<\sqrt{ t }- t ^{3 / 2}+\frac{1}{2} t ^{5 / 2}$ for $t \in(0,1)$
$ \int\limits_{0}^{ x ^{2}} \sqrt{ t } e ^{- t } dt < \int\limits_{0}^{ x ^{2}}\left(\sqrt{ t }- t ^{3 / 2}+\frac{1}{2} t ^{5 / 2}\right) dt$
$=\frac{2}{3} x ^{3}-\frac{2}{5} x ^{5}+\frac{1}{7} x ^{7}$