Claims About a Piecewise Function

Let

Determine, with proof, which of the following claims are true.

1. When $a=1$, $f(x)$ has exactly one zero. 2. For every $a>3$, $f(x)$ has neither a maximum nor a minimum value. 3. There exists a real number $a$ for which $f(x)$ is increasing on $\mathbb R$. 4. If $f(x)$ has a minimum value, then the smallest possible $a$ is $-1$.