Four Claims About a Quadratic Sum

Let

where $a\leqslant b\leqslant c$. Determine, with proof, which of the following claims are true.

1. $f(b)\leqslant 0$. 2. If $b=\dfrac{a+c}2$, then $f(x)\geqslant f(b)$ for all $x\in\mathbb R$. 3. If $b\leqslant \dfrac{a+c}2$, then $f(a)\leqslant f(c)$. 4. $f(a)=f(c)$ holds if and only if $b=0$.