Extremas of a Constrained Inequality!

Algebra Level 5

\[\large{ \dfrac{x^4 + y^4 + z^4}{(x+y+z)^4} }\]

Let the minimum and maximum values of the above expression be \({\alpha}\) and \({\beta},\) respectively, satisfying the following conditions:

  • \(x,y,z \in \mathbb R^+\)
  • \((x+y+z)^3 = 32xyz.\)

Suppose \({\alpha}\) can be represented as \(\dfrac{A - B \sqrt{C}}{D},\) and \({\beta}\) as \(\dfrac{E}{F},\) for positive integers \(A,B,C,D,E,F\) and with \(C\) having no square factors.

Then find the minimum value of \(A+B+C+D+E+F\).

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