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a-b-c-R-and-abc-1-Prove-c-a-3-b-3-a-2-b-2-b-a-3-c-3-a-2-c-2-a-b-3-c-3-b-2-c-2-3-2-




Question Number 176174 by Matica last updated on 14/Sep/22
   a,b,c ∈R_(+ ) ^∗   and abc=1. Prove     ((c (√(a^3 +b^3 )))/(a^2 +b^2 )) + ((b (√(a^3 +c^3 )))/(a^2 +c^2 )) + ((a(√(b^3 +c^3 )))/(b^2 +c^2 )) ≥ (3/( (√2)))
a,b,cR+andabc=1.Proveca3+b3a2+b2+ba3+c3a2+c2+ab3+c3b2+c232
Answered by behi834171 last updated on 14/Sep/22
Σ((c(√(a^3 +b^3 )))/(a^2 +b^2 ))≥3(((abc.Π(a^3 +b^3 ))/(Π(a^2 +b^2 ))))^(1/3) ≥  ≥3(((2(√(a^3 b^3 ))×2(√(b^3 c^3 ))×2(√(c^3 a^3 )))/(2(√(a^2 b^2 ))×2(√(b^2 c^2 ))×2(√(c^2 a^2 )))))^(1/3) ≥  ≥3((abc))^(1/3) =3≥(3/( (√2)))  .■
Σca3+b3a2+b23abc.Π(a3+b3)Π(a2+b2)332a3b3×2b3c3×2c3a32a2b2×2b2c2×2c2a233abc3=332.◼

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