10. The maximum value sought is 2, obtained when $r_{1}=r_{2}=r_{3}=\frac{1}{3}$.
$$
\begin{array}{l}
\text { By } 1+r_{3}=\left(1-r_{1}\right)+\left(1-r_{2}\right) \\
\geqslant 2 \sqrt{\left(1-r_{1}\right)\left(1-r_{2}\right)} \\
\Rightarrow\left(1+r_{1}\right)\left(1+r_{2}\right)\left(1+r_{3}\right) \\
\quad \geqslant 8\left(1-r_{1}\right)\left(1-r_{2}\right)\left(1-r_{3}\right) \\
\Rightarrow 7 a-9 b \leqslant 2 .
\end{array}
$$
10. (20 points) Let $r_{1}, r_{2}, r_{3}$ be the roots of the real-coefficient equation
$$
x^{3}-x^{2}+a x-b=0
$$

with $0<r_{i}<1(i=1,2,3)$. Find the maximum value of $7 a-9 b$.