9. If $a \in \mathbf{R}_{+}, b \in \mathbf{R}$, and
$$
\max \min _{x \in \mathbb{R}}\left\{2 x+4, a x^{2}+b, 5-3 x\right\}=2 \text {, }
$$
then $a+b=(\quad)$.
(A) -1
(B) 1
(C) 2
(D) 3
9. C.

Notice that the lines $y=2x+4$ and $y=5-3x$ pass through the points $(-1,2)$ and $(1,2)$, respectively, and these two points are symmetric with respect to the $y$-axis.

Therefore, when $b=0$, the parabola $y=ax^2$ passes through the points $(-1,2)$ and $(1,2)$, yielding $a=2$;

When $b \neq 0$, the parabola $y=ax^2+b$ passes through the points $(-1,2)$ and $(1,2)$, giving $a(-1)^2 + b = 2$.
In summary, $a+b=2$.