基础不等式

这里将会持续添加一些基础的不等式定理、题目。

Basic Inequalities

Theorem1. 1

$xy\geq2\sqrt{xy}$ . $x,y\in \R ^+$

Proof

\begin{aligned} (x-y)^2&\geq0\\ x^2-2xy+y^2&\geq0\\ x^2+2xy+y^2&\geq4xy\\ (x+y)^2&\geq4xy\\ x+y&\geq2\sqrt{xy}\\ \Leftrightarrow xy&\leq\frac{x+y}{2}\\ Equalitiy\ \ occurs\ \ if\ \ and\ \ only\ \ if\ \ x=y. \end{aligned}

Exercise1. 1

Use Theorem1. 1 to solve this problem.

Let $0<x<4$ . Prove the inequality:

$x(8-2x)\leq8$ .

Solution

\begin{aligned} 2x+(8-2x)=8\\ x(8-2x)=\frac{1}{2}[2x(8-2x)]\leq\frac{1}{2}(\frac{2x-8-2x}{2})^2=8 \end{aligned}

Exercise1.2

Let $n\in R$ . Prove the inequality:

$1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}\leq2$ .

Solution

It’s easy to know:

\frac{1}{a\times a}\leq\frac{1}{a\times(a-1)}.

\begin{aligned} 1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{n^2}<1+\frac{1}{1\times2}+\frac{1}{2\times3}+...+\frac{1}{(n-1)\times n}\\ =1+\frac1 1 -\frac1 2 +\frac1 2-\frac1 3+...+\frac{1}{n-1}-\frac{1}{n}=2-\frac{1}{n}<2 \end{aligned}