JEE Main 2022 (Online) 25th June Morning ShiftQuestion 1
Let $f:\mathbf{R}\to\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be two functions defined by $f(x)=\log_e\left(x^{2}+1\right)-e^{-x}+1$ and $g(x)=\frac{1-2e^{2x}}{e^{x}}$. Then, for which of the following range of $\alpha$, the inequality $f\left(g\left(\frac{(\alpha-1)^{2}}{3}\right)\right)>f\left(g\left(\alpha-\frac{5}{3}\right)\right)$ holds ?
Functions
Solution
$f$ is increasing and $g(x)=e^{-x}-2e^x$ is decreasing, so the inequality means $\frac{(\alpha-1)^2}3<\alpha-\frac53$, i.e. $\alpha^2-5\alpha+6<0$, so $\alpha\in(2,3)$.