# AM-GM inequality

ចំពោះគ្រប់ចំនួនវិជ្ជមាន $x_1 ; x_2 ; ... ; x_n ; 2\leq n\in N$ គេបាន

$\displaystyle \frac{x_1 + x_2 + ... + x_n}{n}\geq \sqrt[n]{x_1x_2...x_n}$

## សំរាយបញ្ជាក់

1. របៀបទី​

តាង $f(x) = lnx , x > 0$

$\displaystyle f'(x) = \frac{1}{x}$

$\displaystyle f''(x) = -\frac{1}{x^2} < 0$

តាម វិសមភាព Jensen ចំពោះ $x_1 ; x_2 ; ... ; x_n\in I ; \forall{2\leq n\in N}$ គេបាន:

$\displaystyle \frac{f(x_1) + f(x_2) + ... + f(x_n)}{n}\leq f(\frac{x_1 + x_2 + ... + x_n}{n})$

$\displaystyle \frac{lnx_1 + lnx_2 + ... + lnx_n}{n}\leq ln(\frac{x_1 + x_2 + ... + x_n}{n})$

$\displaystyle \frac{1}{n}ln(x_1x_2 ... x_n)\leq ln(\frac{x_1 + x_2 + ... + x_n}{n})$

$\displaystyle ln(\sqrt[n]{x_1x_2 . . . x_n})\leq ln(\frac{x_1 + x_2 + . . . + x_n}{n})$

$\displaystyle \Longrightarrow \frac{x_1 + x_2 + ... + x_n}{n}\geq \sqrt[n]{x_1x_2 ... x_n}$

របៀបទី 2

តាង $f(x) = x - 1 - lnx , \forall{x > 0}$

$\displaystyle f'(x) = 1 - \frac{1}{x} = \frac{x - 1}{x}$

$f'(x) = 0 \Longrightarrow x = 1$

• តារាងអថេរភាព

តាមតាងរាងអថេរភាពយើងបាន $f(x)\geq 0 ; \forall{x > 0}$

$\Longrightarrow lnx\leq x - 1$

តាង $\displaystyle t_i = \frac{nx_i}{\sum_{i = 1}^{n}x_i}$

តាមវិសមភាពខាងលើគេបាន

$lnt_i\leq t_i - 1$

$\displaystyle \Longrightarrow \sum_{i = 1}^{n}lnt_i\leq \sum_{i = 1}^{n}t_i - n = 0$

$\Longrightarrow ln(t_1t_2 ... t_n)\leq 0$

$\Longrightarrow t_1t_2 ... t_n\leq 1$

$\displaystyle \Longrightarrow \frac{n^nx_1x_2 ... x_n}{(x_1 + x_2 + ... + x_n)^n}\leq 1$

$\displaystyle \Longrightarrow \frac{x_1 + x_2 + ... + x_n}{n}\geq \sqrt[n]{x_1x_2 ... x_n}$

របៀបទី3 អនុវត្តន៍វិសមភាព Van Khea ចំពោះចំនួនវិជ្ជមាន $x_1, x_2, ..., x_n$ គេបានៈ

$\displaystyle \frac{1^{n+1}}{x_1x_2...x_n}+\frac{1^{n+1}}{x_2...x_nx_1}+...+\frac{1^{n+1}}{x_nx_1...x_{n-1}}\geq \frac{(1+1+...+1)^{n+1}}{(x_1+x_2+...+x_n)(x_2+...+x_n+x_1)...(x_n+x_1+...+x_{n-1})}$

$\displaystyle \Longrightarrow \frac{n}{x_1x_2...x_n}\geq \frac{n^{n+1}}{(x_1+x_2+...+x_n)^n}$

$\Longrightarrow x_1+x_2+...+x_n\geq n\sqrt[n]{x_1x_2...x_n}$

ត្រឡប់មកទំព័រ វិសមភាព វ៉ាន់​ ឃា

បន្តទៅទំព័រ វិសមភាព Bunhiacopski