9. (16 points) Given real numbers $a, b, c$ satisfy: $a+b+c=5, ab+bc+ca=7$.
Find the maximum value of $abc$.

保留了原文的换行和格式。
$$
\begin{array}{l}
b(a+c)+c a=7 \Rightarrow b(5-b)+c a=7 \\
\Rightarrow c a=b^{2}-5 b+7 \leqslant\left(\frac{a+c}{2}\right)^{2}=\left(\frac{5-b}{2}\right)^{2} \\
\Rightarrow \frac{1}{3} \leqslant b \leqslant 3 .
\end{array}
$$

Then $a b c=b\left(b^{2}-5 b+7\right)$
$$
\begin{array}{l}
=b^{3}-5 b^{2}+7 b \\
=(b-1)^{2}(b-3)+3 \\
\leqslant 3 .
\end{array}
$$

When $b=3$, $a b c=3$, at this time, $(a, c)=(1,1)$, which meets the condition;

When $b=1$, $a b c=3$, at this time, $(a, c)=(3,1)$ or $(1,3)$, which meets the condition.
Therefore, the maximum value of $a b c$ is 3.