24. Let the polynomial $x^{3}-a x^{2}+b x-a$ where $(a, b \in \mathbf{R})$ have all real roots. If $a$ is the smallest positive real number that satisfies this condition, and for this real number $a$, the real number $b$ is uniquely determined, then the value of $b$ is ( ).
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12
24. B.
Let $r, s, t$ be the three real roots of the polynomial
$$
x^{3}-a x^{2}+b x-a \quad (a, b \in \mathbf{R})
$$

By Vieta's formulas, we have
$$
\begin{array}{l}
r+s+t=a, \quad rs+st+rt=b, \quad rst=a \\
\Rightarrow r+s+t=rst .
\end{array}
$$

By the AM-GM inequality, we get
$$
\begin{array}{l}
\frac{a}{3}=\frac{r+s+t}{3} \geqslant \sqrt[3]{rst}=\sqrt[3]{a} \\
\Rightarrow a \geqslant 3 \sqrt{3} .
\end{array}
$$

Since $a$ is the smallest positive real number that satisfies the condition, then
$$
\begin{array}{l}
a=3 \sqrt{3} . \\
\text { Hence } r+s+t=3 \sqrt{3}, \quad rst=3 \sqrt{3} \\
\Rightarrow r=s=t=\sqrt{3} \\
\Rightarrow b=rs+st+rt=9 .
\end{array}
$$