A Product Inequality Splits

Suppose that for every $x \in [1, +\infty)$,

Determine, with proof, which of the following are true:

1. if $a \in \left(0, \dfrac{1}{\mathrm{e}}\right)$, then $b \leqslant \mathrm{e}$; 2. if $a \in \left(0, \dfrac{1}{\mathrm{e}}\right)$, then $b > \mathrm{e}$; 3. if $a \in \left[\dfrac{1}{\mathrm{e}}, \mathrm{e}\right)$, then $a^b = \mathrm{e}^{\mathrm{e}}$; 4. if $a \in \left[\dfrac{1}{\mathrm{e}}, \mathrm{e}\right)$, then $b^a = \mathrm{e}^{\mathrm{e}}$.