5. The maximum value of the function $f(x)=(6-x)^{3}(x-1)^{\frac{2}{3}}(1 \leqslant x \leqslant 6)$ is

保留了源文本的换行和格式。
$$
\begin{array}{l}
\text { 5. } \frac{3^{6} \times 5^{3}}{11^{4}} \sqrt[3]{1100} \\
(f(x))^{3}=(6-x)^{9}(x-1)^{2} \\
=\frac{1}{2^{9}}(12-2 x)^{9} \cdot \frac{1}{9^{2}}(9 x-9)^{2} \\
\leqslant \frac{1}{2^{9} \times 9^{2}}\left[\frac{9(12-2 x)+2(9 x-9)}{11}\right]^{11} \\
=\frac{9^{11} \times 2^{11}}{2^{9} \times 9^{2}}\left(\frac{5}{11}\right)^{11}=2^{2} \times 3^{18}\left(\frac{5}{11}\right)^{11} .
\end{array}
$$

When and only when $12-2 x=9 x-9$, i.e., $x=\frac{21}{11}$,
the equality holds.
$$
\begin{array}{l}
\text { Therefore, } f(x)_{\max }=\sqrt[3]{2^{2} \times 3^{18}\left(\frac{5}{11}\right)^{12} \times \frac{11}{5}} \\
=\frac{3^{6} \times 5^{3}}{11^{4}} \sqrt[3]{1100} .
\end{array}
$$