不等式专项
作者：互联网
1. 重要不等式
\(a^2+b^2\ge 2ab\)
2. 基本不等式
\(a\ge0,b\ge0,\dfrac{a+b}{2}\ge\sqrt{ab}\)
3. 均值不等式
\(\dfrac{2}{\frac1a+\frac1b}\le\sqrt{ab}\le\dfrac{a+b}{2}\le\sqrt{\dfrac{a^2+b^2}{2}}\)
当且仅当 \(a=b\) 时等号成立。
拓展：
\(\text{调和均值：}H_{n}=\dfrac{n}{\sum \limits_{i=1}^{n}\frac{1}{x_{i}}}= \dfrac{n}{\frac{1}{x_{1}}+ \frac{1}{x_{2}}+ \cdots + \frac{1}{x_{n}}} \\\)
\(\text{几何均值：}G_{n}=\sqrt[n]{\prod \limits_{i=1}^{n}x_{i}}= \sqrt[n]{x_{1}x_{2}\cdots x_{n}} \\\)
\(\text{算数均值：}A_{n}=\dfrac{1}{n}\sum \limits_{i=1}^{n}x_{i}=\dfrac{x_{1}+ x_{2}+ \cdots + x_{n}}{n} \\\)
\(\text{平方均值：}Q_{n}=\sqrt{\dfrac{1}{n} \sum \limits_{i=1}^{n}x_{i}^{2}}= \sqrt{\dfrac{x_{1}^{2}+ x_{2}^{2}+ \cdots + x_{n}^{2}}{n}} \\\)
\(H_{n}\leq G_{n}\leq A_{n}\leq Q_{n}\)
标签：专项,frac,limits,不等式,dfrac,sqrt,cdots,text 来源： https://www.cnblogs.com/adm-1223/p/16486665.html