Ta có:
\(\frac{2.\left(x^2+x+1\right)}{x^2+1}=\frac{2.\left(x^2+1\right)+2x}{x^2+1}=2+\frac{2x}{x^2+1}\)
Ta có:\(2+\frac{2x}{x^2+1}-1=1+\frac{2x}{x^2+1}\)
\(=\frac{x^2+2x+1}{x^2+1}=\frac{\left(x+1\right)^2}{x^2+1}\ge0\) \(\Rightarrow\frac{2.\left(x^2+x+1\right)}{x^2+1}\ge1\)
\(2+\frac{2x}{x^2+1}-3=\frac{2x}{x^2+1}-1=\frac{-x^2+2x-1}{x^2+1}\)
\(=\frac{-\left(x-1\right)^2}{x^2+1}\le0\) \(\Rightarrow\frac{2.\left(x^2+x+1\right)}{x^2+1}\le3\)
Vậy \(1\le\frac{2.\left(x^2+x+1\right)}{x^2+1}\le3\)
CMR: \(\frac{1}{3}\le\frac{x^2+x+1}{x^2-x+1}\le3\)
Cho x,y là các số thực
CMR: \(-\frac{1}{4}\le\frac{\left(x^2-y^2\right)\left(1x^2y^2\right)}{\left(1+x^2\right)\left(1+y^2\right)^2}\le\frac{1}{4}\)
Cho x,y,z ko âm t/m xyz=1. CMR
\(\frac{1}{\left(x+1\right)^2+y^2+1}+\frac{1}{\left(y+1\right)^2+z^2+1}+\frac{1}{\left(z+1\right)^2+x^2+1}\le\frac{1}{2}\)
1.Cho a,b,c dương. CMR: \(\left(x+y+z\right)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\ge9\)
2. Cho a,b khác 0. CMR: \(a^4+b^4\le3\left(\frac{a^6}{b^2}+\frac{b^6}{a^2}\right)\)
CÁC THIÊN TÀI ĐÂU!!! Giúp với nhé!
\(\frac{1}{3}\le\frac{^{x^2}+x+1}{x^2-x+1}\le3\)CM
gần gấp!!!!!!
Cho \(x,y,z>0\)
CMR: \(\frac{x}{z^2+y^2}+\frac{y}{x^2+y^2}+\frac{z}{x^2+y^2}\le\frac{1}{2}\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)\)
1.Cho a,b,c là 3 số dương. Chứng minh :
a) \(\frac{a+1}{b+2c+3}+\frac{b+1}{c+2a+3}+\frac{c+1}{a+2b+3}\ge1\)
b) \(\sqrt{\frac{a}{7a^2+4}}+\sqrt{\frac{a}{7b^2+4}}+\sqrt{\frac{a}{7c^2+4}}\le27\left(\frac{1}{42a+29}+\frac{1}{42b+29}+\frac{1}{42c+29}\right)\)
c) \(c^2-a^2-b^2\le4\left(ĐK:2\le c\le3;\frac{b}{2}+\frac{3}{c}\ge2;a+\frac{b}{2}+\frac{c}{3}\ge3\right)\)
2. Chứng minh :
a) \(2x+\sqrt{12-2x^2}\le6\left(ĐK:6-x^2\ge0\right)\)
b) \(\sqrt{1-2y-y^2}\le y+3\left(ĐK:1-2y-y^2\ge0\right)\)
c) \(\sqrt{5-x^2}+\sqrt{5-\frac{1}{x^2}}+x+\frac{1}{x}\ge6\left(ĐK:5-x^2\ge0;5-\frac{1}{x^2}\ge0\right)\)
Bài 1: Cho a,b,c đôi một khác nhau. CMR:
\(\frac{\left(x-b\right)\left(x-c\right)}{\left(a-b\right)\left(a-c\right)}+\frac{\left(x-c\right)\left(x-a\right)}{\left(b-c\right)\left(b-a\right)}+\frac{\left(x-a\right)\left(x-b\right)}{\left(c-a\right)\left(c-b\right)}=1\)=1
Bài 2: CMR: nếu \(\frac{1}{x}-\frac{1}{y}-\frac{1}{z}=1\)và x=y+z thì:
\(\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2}=1\)
Cho \(x^2+4y^2=1\)
\(CMR:\left|x-y\right|\le\frac{\sqrt{5}}{2}\)
