Prove that for every positive real numbers x, y, z and xyz = 1, hold (x + y + z)^2 ((1/x^2 ) + (1/y^2 ) + (1/z^2 )) ≥ 9 + 2(x^3 + y^3 + z^3 ) + 4((1/x^3 ) + (1/y^3 ) + (1/z^3 ))


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Question Number 34031 by naka3546 last updated on 29/Apr/18

$P r o v e t h a t f o r e v e r y p o s i t i v e r e a l n u m b e r s x, y, z a n d x y z = 1, h o l d$ ${(x + y + z)}^{2} (\frac{1}{x^{2}} + \frac{1}{y^{2}} + \frac{1}{z^{2}}) ⩾ 9 + 2 (x^{3} + y^{3} + z^{3}) + 4 (\frac{1}{x^{3}} + \frac{1}{y^{3}} + \frac{1}{z^{3}})$
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