Comprehension

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 \(g(x)=\int\limits_{0}^{x^{2}} \sqrt{t} e^{-t} d t, \)\(x > 0\) .

Question: 1

Which of the following statements is TRUE ?

Updated On: May 22, 2024
  • $f(\sqrt{\ln 3})+g(\sqrt{\ln 3})=\frac{1}{3}$
  • For every $x > 1$, there exists and $\alpha \in(1, x)$ such that $\psi_{1}(x)=1+\alpha x$
  • For every $x > 0$, there exists a $\beta \in(0, x)$ such that $\psi_{2}(x)=2 x\left(\psi_{1}(\beta)-1\right)$
  • $f$ is an increasing function on the interval $\left[0, \frac{3}{2}\right]$
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The Correct Option is C

Solution and Explanation

The correct answer is (C): For every $x > 0$, there exists a $\beta \in(0, x)$ such that $\psi_{2}(x)=2 x\left(\psi_{1}(\beta)-1\right)$

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Question: 2

Which of the following statements is TRUE?

Updated On: Aug 19, 2024
  • $\psi_{1}( x ) \leq 1$, for all $x >0$
  • $\psi_{2}( x ) \leq 0$, for all $x >0$
  • $f(x) \geq 1-e^{-x^{2}}-\frac{2}{3} x^{3}+\frac{2}{5} x^{5}$, for all $x \in\left(0, \frac{1}{2}\right)$
  • $g(x) \leq \frac{2}{3} x^{3}-\frac{2}{5} x^{5}+\frac{1}{7} x^{7}$, for all $x \in\left(0, \frac{1}{2}\right)$
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The Correct Option is D

Solution and Explanation

The correct option is (D): $g(x) \leq \frac{2}{3} x^{3}-\frac{2}{5} x^{5}+\frac{1}{7} x^{7}$, for all $x \in\left(0, \frac{1}{2}\right)$

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