15. (16 points) Let $x_{1}, x_{2}, x_{3} \in [0,12]$,
$$
x_{1} x_{2} x_{3}=\left(\left(12-x_{1}\right)\left(12-x_{2}\right)\left(12-x_{3}\right)\right)^{2} \text {. }
$$

Find the maximum value of $f=x_{1} x_{2} x_{3}$.
$$
\text { III. 15. Let } t=\sqrt[6]{x_{1} x_{2} x_{3}} \geqslant 0 \text {. }
$$

Then by the problem statement,
$$
\begin{aligned}
t & =\sqrt[6]{x_{1} x_{2} x_{3}} \\
& =\sqrt[3]{\left(12-x_{1}\right)\left(12-x_{2}\right)\left(12-x_{3}\right)} . \\
& \leqslant \frac{\left(12-x_{1}\right)+\left(12-x_{2}\right)+\left(12-x_{3}\right)}{3} \\
& =12-\frac{x_{1}+x_{2}+x_{3}}{3} \\
& \leqslant 12-\sqrt[3]{x_{1} x_{2} x_{3}}=12-t^{2} .
\end{aligned}
$$
$$
\text { Hence } t^{2}+t-12 \leqslant 0 \Rightarrow 0 \leqslant t \leqslant 3 \text {. }
$$

Equality holds if and only if $x_{1}=x_{2}=x_{3}=9$.
Thus, $f=x_{1} x_{2} x_{3}=t^{6} \leqslant 3^{6}=729$.
Therefore, $f_{\max }=729$.