Inegalitate cu variabile subunitare
Posted: Tue Jan 27, 2009 11:06 am
Sa se demonstreze ca, daca \( a,b,c\in [0,1] \) atunci: \( \frac{1}{5-ab}+\frac{1}{5-bc}+\frac{1}{5-ca} \ge \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{4} \).
Se arata ca
\( \frac{1}{5-bc}\ge \frac{\sqrt{b}+\sqrt{c}}{8} \)
\( \frac{1}{5-ca}\ge \frac{\sqrt{c}+\sqrt{a}}{8} \)
\( \frac{1}{5-ab}\ge \frac{\sqrt{a}+\sqrt{b}}{8} \) si apoi se aduna.
Intr-adevar \( (1-\sqrt{b})(1-\sqrt{c})\ge 0 \) deci \( \sqrt{b}+\sqrt{c}\le 1+\sqrt{bc} \)
si atunci \( \frac{\sqrt{b}+\sqrt{c}}{8}\le\frac{1+\sqrt{bc}}{8}\le \frac{1}{5-bc}\Longleftrightarrow (\sqrt{bc}-1)^2(\sqrt{bc}+3)\ge 0 \)