QJEEM22Mabc50: JEE Main 2022 Mathematics - Class 11 JEEM22Mabc50 Solution: Let and be two functions defined by and . Then, for which of…

$f(x) = {\log _e}({x^2} + 1) - {e^{ - x}} + 1$

$f'(x) = {{2x} \over {{x^2} + 1}} + {e^{ - x}}$

= {2 \over {x + {1 \over x}}} + {e^{ - x}} > 0\,\,\forall x \in R

$g(x) = {e^{ - x}} - 2{e^x}$

g'(x) - - {e^{ - x}} - 2{e^x} < 0\,\,\,\,\forall x \in R

$\Rightarrow$ f(x) is increasing and g(x) is decreasing function.

f\left( {g\left( {{{{{(\alpha - 1)}^2}} \over 3}} \right)} \right) > f\left( {g\left( {\alpha - {5 \over 3}} \right)} \right)

\Rightarrow {{{{(\alpha - 1)}^2}} \over 3} < \alpha - {5 \over 3}

= {\alpha ^2} - 5\alpha + 6 < 0

= (\alpha - 2)(\alpha - 3) < 0

$= \alpha \in (2,\,3)$