Determine the greatest value of the following expression

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The discussion focuses on the mathematical problem of determining the greatest value of the expression $$P=\frac{11a^2-3\sqrt3ab-\frac{1}{3}}{9b-10\left ( \sqrt3a-1 \right )}$$ given the cubic equation $ax^3-x^2+ax-b=0$ with three positive real roots. Participants inquire about the course context of the problem to facilitate targeted assistance. The conversation emphasizes the importance of understanding the problem's origin for effective problem-solving.

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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:

$$P=\frac{11a^2-3\sqrt3ab-\frac{1}{3}}{9b-10\left ( \sqrt3a-1 \right )}$$
 
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From what course is this problem taken?

What progress have you made so far?
 
simple, i love inequality!(h)
 
I asked from what course the problem is taken, so that I could determine if this sub-forum is the best choice.

I asked what you progress so far has been so that our helpers will know just where you are stuck, and be able to provide meaningful help.

Your response addresses neither question.
 
Thank
But I don't have imagine with my problem
 
Was this problem assigned to you by a teacher/professor in a class?

If so, what is the name of the class?
 

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