A Piecewise Polynomial Ladder

A function $f(x)$ on $(0, +\infty)$ is defined piecewise: for $n - 1 < x \leqslant n$ (with $n \in \mathbb{N}^*$),

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

1. $f(x)f(x+1) \leqslant 0$ for all $x > 0$; 2. for $t > 0$: $f(x)$ has exactly two zeros in $(t, 2t)$ if and only if $t \in \left(\dfrac{3}{2}, \dfrac{5}{2}\right)$; 3. there exist positive reals $a$ and $x_0$ such that $f(x) < e^{-ax}$ for all $x > x_0$; 4. for $2 \leqslant t < 5$: $f(x)$ has exactly two critical points in $(2t - 4, t + 1)$ if and only if $t \in \left[\dfrac{8}{3}, \dfrac{25}{8}\right) \cup \left(\dfrac{19}{6}, \dfrac{18}{5}\right)$.