if-x-y-z-gt-0-and-xyz-8-prove-that-1-x-2-8-1-y-2-8-1-z-2-8-1-4-

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

i f x; y; z > 0 a n d x y z = 8 p r o v e t h a t :

\frac{1}{x^{2} + 8} + \frac{1}{y^{2} + 8} + \frac{1}{z^{2} + 8} ⩽ \frac{1}{4}

Answered by dumitrel last updated on 02/Aug/21

p^{3} ⩾ 27 r \Rightarrow p ⩾ 6

q^{2} ⩾ 3 p r \Rightarrow q ⩾ 12

Σ \frac{1}{x^{2} + 8} = \frac{3}{8} + Σ (\frac{1}{x^{2} + 8} - \frac{1}{8}) = \frac{3}{8} - \frac{1}{8} Σ \frac{x^{2}}{x^{2} + 8} ⩽

⩽ \frac{3}{8} - \frac{1}{8} \cdot \frac{p^{2}}{p^{2} - 2 q + 24} \overset{∙}{⩽} \frac{1}{4}

∙ \Leftrightarrow \frac{p^{2}}{p^{2} - 2 q + 24} ⩾ 1 \Leftrightarrow q ⩾ 12

Commented by mathdanisur last updated on 02/Aug/21

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