Inegalitate utila....
Posted: Sat Apr 19, 2008 2:47 pm
Fie \( x_1,x_2,...,x_n\geq0 \) cu \( x_1\cdot...\cdot x_n=1 \). Demonstrati inegalitiatea:
\( \sum x_i^{k+1}\geq\sum x_i^{k} \)
\( \sum x_i^{k+1}\geq\sum x_i^{k} \)
\( x_1x_2...x_n=1 \Rightarrow \frac {1}{n} (\sum_{i=1}^{n} {x_i}) \geq 1 \)
Acum avem: \( \sum_{i=1}^{n} {{x_i}^{k+1}} \) \( \geq \) \( \frac {1}{n} (\sum_{i=1}^{n} {x_i})(\sum_{i=1}^{n} {{x_i}^k}) \) \( \geq \) \( \sum_{i=1}^{n} {{x_i}^k} \) QED