Example 3 Given $x \in \mathbf{R}$, let $M=\max \{1-x, 2 x-$ $3, x\}$. Find $M_{\min }$.
Solution: Since $M$ is the maximum of $1-x, 2x-3, x$, $M$ is not less than the maximum of $1-x, x$, and thus $M$ is not less than the arithmetic mean of $1-x$ and $x$, i.e.,
$$
M \geqslant \frac{(1-x)+x}{2}=\frac{1}{2} \text {. }
$$

When $x=\frac{1}{2}$, equality holds, so $M_{\text {min }}=\frac{1}{2}$.