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Solve-x-y-z-R-amp-x-y-z-K-Min-x-4-y-4-z-4-x-2-y-2-Source-Elementary-Ol




Question Number 153340 by mnjuly1970 last updated on 06/Sep/21
       Solve ..                    x , y , z ∈ R^( +)  &  x+ y= z                    K := Min_   ((( x^( 4)  + y^( 4) + z^( 4) )/(x^( 2) y^( 2) )) ) = ?          ■               Source :  Elementary Olympid Book  m.n
$$ \\ $$$$\:\:\:\:\:\mathrm{Solve}\:.. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{x}\:,\:\mathrm{y}\:,\:\mathrm{z}\:\in\:\mathbb{R}^{\:+} \:\&\:\:\mathrm{x}+\:\mathrm{y}=\:\mathrm{z} \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{K}\::=\:\mathrm{Min}_{\:} \:\left(\frac{\:\mathrm{x}^{\:\mathrm{4}} \:+\:\mathrm{y}^{\:\mathrm{4}} +\:\mathrm{z}^{\:\mathrm{4}} }{\mathrm{x}^{\:\mathrm{2}} \mathrm{y}^{\:\mathrm{2}} }\:\right)\:=\:?\:\:\:\:\:\:\:\:\:\:\blacksquare\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\mathrm{Source}\::\:\:\mathrm{Elementary}\:\mathrm{Olympid}\:\mathrm{Book}\:\:{m}.{n} \\ $$$$ \\ $$
Answered by mr W last updated on 06/Sep/21
symmetry ⇒x=y=(z/2)  ⇒min=18
$${symmetry}\:\Rightarrow{x}={y}=\frac{{z}}{\mathrm{2}} \\ $$$$\Rightarrow{min}=\mathrm{18} \\ $$
Commented by mnjuly1970 last updated on 06/Sep/21
  thanks alot mr W ...
$$\:\:{thanks}\:{alot}\:{mr}\:{W}\:… \\ $$

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