回復 38# shihtc 的帖子
第4題
\(\begin{align}
& {{x}_{1}}+x{}_{2}+\cdots +{{x}_{8}}=4 \\
& {{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+\cdots +{{x}_{7}}{{x}_{8}}=7 \\
& \\
& {{x}_{1}}^{2}+x{{{}_{2}}^{2}}+\cdots +{{x}_{8}}^{2} \\
& ={{\left( {{x}_{1}}+x{}_{2}+\cdots +{{x}_{8}} \right)}^{2}}-2\left( {{x}_{1}}{{x}_{2}}+{{x}_{1}}{{x}_{3}}+\cdots +{{x}_{7}}{{x}_{8}} \right) \\
& =2 \\
& \\
& {{\left( {{x}_{1}}+x{}_{2}+\cdots +{{x}_{8}} \right)}^{2}}\le \left( {{x}_{1}}^{2}+x{{{}_{2}}^{2}}+\cdots +{{x}_{8}}^{2} \right)\left( {{1}^{2}}+{{1}^{2}}+\cdots +{{1}^{2}} \right) \\
\end{align}\)
等號成立於\({{x}_{1}}={{x}_{2}}=\cdots ={{x}_{8}}=\frac{1}{2}\)