AMC 12 2016a Problem 24

There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is the value of $b?$

\textbf{(A)}\ 8 \textbf{(B)}\ 9 \textbf{(C)}\ 10 \textbf{(D)}\ 11 \textbf{(E)}\ 12