MHB Finding Min Value of $\dfrac{|b|+|c|}{a}$ from Roots of Cubic Equations

[anemone]

If $\alpha,\,\beta,\,\gamma$ are the roots of the equation $x^3+ax+1=0$, where $a$ is a positive real number and $\dfrac{\alpha}{\beta},\,\dfrac{\beta}{\gamma},\,\dfrac{\gamma}{\alpha}$ be the roots of the equation $x^3+bx^2+cx-1=0$, find the minimum value of $\dfrac{|b|+|c|}{a}$.

 

[solakis1]

is the answer 3/a

If $\alpha,\,\beta,\,\gamma$ are the roots of the equation $x^3+ax+1=0$, where $a$ is a positive real number and $\dfrac{\alpha}{\beta},\,\dfrac{\beta}{\gamma},\,\dfrac{\gamma}{\alpha}$ be the roots of the equation $x^3+bx^2+cx-1=0$, find the minimum value of $\dfrac{|b|+|c|}{a}$.

Is the answer $3/a$

 

[anemone]

Is the answer $3/a$

Nope, sorry solakis!

 

[solakis1]

is the answer a number?

 

[anemone]

Hi solakis and to all MHB members,

I am sorry that I still haven't gotten around to follow up all my unanswered challenges here in MHB! (Sadface) I promise that once I got my personal things straighten out a bit more, I will have more time for MHB then.

To answer to your query, solakis, the answer to this problem is $3888^{\tiny\dfrac{1}{6}}$.