# 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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