6. For three numbers $a, b, c$, use $\min \{a, b, c\}$ to represent the smallest of these three numbers, for example,
$$
\begin{array}{l}
\min \{-1,2,3\}=-1, \\
\min \{-1,2, a\}=\left\{\begin{array}{cc}
a, & a \leqslant-1 ; \\
-1, & a>-1 .
\end{array}\right.
\end{array}
$$

Then, the maximum value of $\min \left\{x+1,(x-1)^{2}, 2-x\right\}$ is ( ).
(A) $x+1$
(B) $(x-1)^{2}$
(C) $2-x$
(D) 1
6. D.

Draw the graphs of $y=x+1$, $y=(x-1)^{2}$, and $y=2-x$ as shown in Figure 4.

From the graphs, we know that the maximum value of $\min \left\{x+1,(x-1)^{2}, 2-x\right\}$ is 1.