List 5. Try to find the maximum value of $f(x)=\min \{4 x+1, x+2, -2 x+4\}(x \in R)$.


Solve: In the same Cartesian coordinate system, draw the graphs of the functions
$$
\begin{array}{l}
y_{1}=4 x+1, \\
y_{1}=x+2, \\
y_{3}=-2 x+4 \text { . }
\end{array}
$$

Since for each $x$ value, $f(x)$ is the minimum of $y_{1}, y_{2}, y_{3}$, we have
$$
f(x)=\left\{\begin{array}{ll}
4 x+1, & x<\frac{1}{3}, \\
x+2, & \frac{1}{3} \leq x \leq \frac{2}{3}, \\
-2 x+4, & x>\frac{2}{3} .
\end{array}\right.
$$

Thus, the graph of $y=f(x)$ is shown by the bold line in the figure, and the point $\left(\frac{2}{3}, \frac{8}{3}\right)$ is the highest point on the graph. Therefore, when $x=\frac{2}{3}$, $f(x)$ has the maximum value $\frac{8}{3}$.