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Question Number 162788 by HongKing last updated on 01/Jan/22
let  x;y;z > 0  such that  x^4 +y^4 +z^4  = x^2 +y^2 +z^2   find the minimum of the expression:  P = (x^2 /y) + (y^2 /z) + (z^2 /x)
$$\mathrm{let}\:\:\mathrm{x};\mathrm{y};\mathrm{z}\:>\:\mathrm{0} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\mathrm{x}^{\mathrm{4}} +\mathrm{y}^{\mathrm{4}} +\mathrm{z}^{\mathrm{4}} \:=\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{minimum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{expression}: \\ $$$$\mathrm{P}\:=\:\frac{\mathrm{x}^{\mathrm{2}} }{\mathrm{y}}\:+\:\frac{\mathrm{y}^{\mathrm{2}} }{\mathrm{z}}\:+\:\frac{\mathrm{z}^{\mathrm{2}} }{\mathrm{x}} \\ $$
Answered by alephzero last updated on 01/Jan/22
x^4  + y^4  + z^4  = x^2  + y^2  + z^2   (x_1 ; y_1 ; z_1 ) = (−1; −1; −1)  (x_2 ; y_2 ; z_2 ) = (0; 0; 0)  (x_3 ; y_3 ; z_3 ) = (1; 1; 1)  But x; y; z > 0  ⇒ (x; y; z) = (1; 1; 1)  P = (x^2 /y) + (y^2 /z) + (z^2 /x) = (1^2 /1) + (1^2 /1) + (1^2 /1) = 1 + 1 +  + 1 = 3  P = 3
$${x}^{\mathrm{4}} \:+\:{y}^{\mathrm{4}} \:+\:{z}^{\mathrm{4}} \:=\:{x}^{\mathrm{2}} \:+\:{y}^{\mathrm{2}} \:+\:{z}^{\mathrm{2}} \\ $$$$\left({x}_{\mathrm{1}} ;\:{y}_{\mathrm{1}} ;\:{z}_{\mathrm{1}} \right)\:=\:\left(−\mathrm{1};\:−\mathrm{1};\:−\mathrm{1}\right) \\ $$$$\left({x}_{\mathrm{2}} ;\:{y}_{\mathrm{2}} ;\:{z}_{\mathrm{2}} \right)\:=\:\left(\mathrm{0};\:\mathrm{0};\:\mathrm{0}\right) \\ $$$$\left({x}_{\mathrm{3}} ;\:{y}_{\mathrm{3}} ;\:{z}_{\mathrm{3}} \right)\:=\:\left(\mathrm{1};\:\mathrm{1};\:\mathrm{1}\right) \\ $$$$\mathrm{But}\:{x};\:{y};\:{z}\:>\:\mathrm{0} \\ $$$$\Rightarrow\:\left({x};\:{y};\:{z}\right)\:=\:\left(\mathrm{1};\:\mathrm{1};\:\mathrm{1}\right) \\ $$$$\mathrm{P}\:=\:\frac{{x}^{\mathrm{2}} }{{y}}\:+\:\frac{{y}^{\mathrm{2}} }{{z}}\:+\:\frac{{z}^{\mathrm{2}} }{{x}}\:=\:\frac{\mathrm{1}^{\mathrm{2}} }{\mathrm{1}}\:+\:\frac{\mathrm{1}^{\mathrm{2}} }{\mathrm{1}}\:+\:\frac{\mathrm{1}^{\mathrm{2}} }{\mathrm{1}}\:=\:\mathrm{1}\:+\:\mathrm{1}\:+ \\ $$$$+\:\mathrm{1}\:=\:\mathrm{3} \\ $$$$\mathrm{P}\:=\:\mathrm{3} \\ $$
Commented by HongKing last updated on 01/Jan/22
My dear Sir thank you, but  need to prove  P≥3  for all  x;y;z  such that  x^4 +y^4 +z^4  = x^2 +y^2 +z^2
$$\mathrm{My}\:\mathrm{dear}\:\mathrm{Sir}\:\mathrm{thank}\:\mathrm{you},\:\mathrm{but} \\ $$$$\mathrm{need}\:\mathrm{to}\:\mathrm{prove}\:\:\mathrm{P}\geqslant\mathrm{3}\:\:\mathrm{for}\:\mathrm{all}\:\:\mathrm{x};\mathrm{y};\mathrm{z} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\mathrm{x}^{\mathrm{4}} +\mathrm{y}^{\mathrm{4}} +\mathrm{z}^{\mathrm{4}} \:=\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \\ $$

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