2. Given the function $f(x)=\left|x^{2}-2 a x+b\right|(x \in \mathbf{R})$, the following propositions are given: (1) $f(x)$ must be an even function; (2) When $f(0)=f(2)$, the graph of $f(x)$ is symmetric about the line $x=1$; (3) If $a^{2}-b^{2} \leqslant 0$, then $f(x)$ is an increasing function on the interval $[a,+\infty)$; (4) $f(x)$ has a maximum value $\left|a^{2}-b\right|$. The number of correct propositions is
A. 1
B. 2
C. 3
D. 4
2. $\mathrm{A}$

Only (3) is correct.