TITLE: Find the maximum possible value.
QUESTION [6 upvotes]: For all ordered triples $(p,q,r)$ define the polynomial
$$f_{p,q,r}(x)=x^3-px^2+qx-r$$
Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such that the roots of $f_{a_{1},a_{2},a_{3}}(x)$ are $b_{1},b_{2},b_{3}$ and the roots of $f_{b_{1},b_{2},b_{3}}(x)$ are $c_{1},c_{2},c_{3}$. Determine the maximum possible value of   
$$ \frac{9\sqrt[3]{b_{3}}}{b_{1}+3} + \frac{4+3b_{1}+2b_{2}+b_{3}}{a_{1}+1} $$  

I used Vieta's formulas combined with calculus. I set this expression equal to $y$ and then cubed both sides. Then I tried to use the fact that since $y$ is real, the cubic in $y$ (generated by cubing both sides) will have three real roots. Now, I differentiated the equation w.r.t. $y$ (assuming everything else to be constant). I got a quadratic in $y$ and I then made its discriminant $>0$. Now I used Vieta's formulas. After that I'm stuck since I still have more than one variable left. Also, I'm not yet familiar with multivariable calculus. Any help will be greately appreciated.
  Thanks!

REPLY [4 votes]: This question is an old problem posed on Brilliant.
You can view it here, along with the solution by the problem creator Zi Song.