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Given that the equation $ax^3-x^2+ax-b=0$, ($a\neq 0,\,b\neq 0$) has three positive real roots.

Determine the greatest value of the following expression:

\(\displaystyle P=\frac{11a^2-3\sqrt3ab-\frac{1}{3}}{9b-10\left ( \sqrt3a-1 \right )}\)

Determine the greatest value of the following expression:

\(\displaystyle P=\frac{11a^2-3\sqrt3ab-\frac{1}{3}}{9b-10\left ( \sqrt3a-1 \right )}\)

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