Extremas of a Constrained Inequality!

x4+y4+z4(x+y+z)4\large{ \dfrac{x^4 + y^4 + z^4}{(x+y+z)^4} }

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

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

Suppose α{\alpha} can be represented as ABCD,\dfrac{A - B \sqrt{C}}{D}, and β{\beta} as EF,\dfrac{E}{F}, for positive integers A,B,C,D,E,FA,B,C,D,E,F and with CC having no square factors.

Then find the minimum value of A+B+C+D+E+FA+B+C+D+E+F.


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