Question Number 149036 by mathdanisur last updated on 02/Aug/21

if x;y;z>0 and xyz=8 prove that: (1/(x^2 + 8)) + (1/(y^2 + 8)) + (1/(z^2 + 8)) ≤ (1/4)

Answered by dumitrel last updated on 02/Aug/21

p^3 ≥27r⇒p≥6 q^2 ≥3pr⇒q≥12 Σ(1/(x^2 +8))=(3/8)+Σ((1/(x^2 +8))−(1/8))=(3/8)−(1/8)Σ(x^2 /(x^2 +8))≤ ≤(3/8)−(1/8)∙(p^2 /(p^2 −2q+24))≤^• (1/4) •⇔(p^2 /(p^2 −2q+24))≥1⇔q≥12

Commented bymathdanisur last updated on 02/Aug/21

Cool, Thank You Ser