\begin{aligned} & \text { Example } 5 \max _{a, b, c \in \mathbf{R}_{+}} \min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\} \\ = & \end{aligned}
Solution: Let $t=\min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\}$. Then $0<t \leqslant \frac{1}{a}, 0<t \leqslant \frac{1}{b^{2}}, 0<t \leqslant \frac{1}{c^{3}}$,

which means $a \leqslant \frac{1}{t}, b^{2} \leqslant \frac{1}{t}, c^{3} \leqslant \frac{1}{t}$.
Therefore, $t \leqslant a+b^{2}+c^{3} \leqslant \frac{3}{t}$.
Thus, we get $t \leqslant \sqrt{3}$, and when $a=b^{2}=c^{3}=\frac{\sqrt{3}}{3}$, $t=\sqrt{3}$. Hence,
$$
\max _{a, b, c \in \mathbf{R}_{+}} \min \left\{\frac{1}{a}, \frac{1}{b^{2}}, \frac{1}{c^{3}}, a+b^{2}+c^{3}\right\}=\sqrt{3} \text {. }
$$

Comment: The key is $0<t \leqslant \frac{1}{a}, 0<t \leqslant \frac{1}{b^{2}}, 0<t \leqslant \frac{1}{c^{3}}, 0<t \leqslant a+b^{2}+c^{3}$, then eliminate the variables $a$, $b$, and $c$.