Cho a,b,c>0; có a+b+c\(\le\)3.
Chứng minh rằng:
\(\frac{a}{\sqrt{2a^2+b^2}+\sqrt{3}}+\frac{b}{\sqrt{2b^2+c^2}+\sqrt{3}}+\frac{c}{\sqrt{2c^2+a^2}+\sqrt{3}}\le\frac{\sqrt{3}}{2}\)
Cho a;b;c > 0.Chứng minh \(\sqrt{\frac{a}{b+c+2a}}+\sqrt{\frac{b}{c+a+2b}}+\sqrt{\frac{c}{a+b+2c}}\le\frac{3}{2}\)
1) cho a;b;c ko âm .chứng minh \(\sqrt{\frac{a+2b}{3}}+\sqrt{\frac{b+2c}{3}}+\sqrt{\frac{c+2a}{3}}\ge\sqrt{a}+\sqrt{b}+\sqrt{c}\)
2) cho a;;b;c dương và abc=1. chứng minh \(\frac{bc}{a^2b+a^2c}+\frac{ca}{b^2c+b^2a}+\frac{ab}{c^2a+c^2b}\ge\frac{3}{2}\)
Bài 1: \(\hept{\begin{cases}a,b,c>0\\ab+bc+ca=5abc\end{cases}CMR:P=\frac{1}{2a+2b+c}+\frac{1}{a+2b+2c}+\frac{1}{2a+b+2c}\le}1\)
Bài 2:\(\hept{\begin{cases}a,b,c>0\\a+b+c=9\end{cases}}\)Tìm GTNN \(P=\frac{1}{\sqrt[3]{a+2b}}+\frac{1}{\sqrt[3]{b+2c}}+\frac{1}{\sqrt[3]{c+2a}}\)
Cho các số thực dương a,b,c. CM: \(\sqrt{\frac{a}{b+c+2a}}+\sqrt{\frac{b}{a+c+2b}}\sqrt{\frac{c}{a+b+2c}}\le\frac{3}{2}\)
Cho a+b+c=3 và a,b,c>0. Tìm Min A=\(\frac{a\sqrt{a}}{\sqrt{a+b+2c}}+\frac{b\sqrt{b}}{\sqrt{b+c+2a}}+\frac{c\sqrt{c}}{\sqrt{c+a+2b}}\)
Cho 3 số thực dương a, b,c CMR \(\sqrt{\frac{2a}{a+b}}+\sqrt{\frac{2b}{b+c}}+\sqrt{\frac{2c}{c+a}}\le3\)\(\le\)3
Cho a,b,c>0. Chứng minh: \(\sqrt{\frac{a}{a+2b}}+\sqrt{\frac{b}{b+2c}}+\sqrt{\frac{c}{c+2a}}>1\)
Cho a,b,c>0 CMR
\(a\sqrt{\frac{a}{a+2b}}+b\sqrt{\frac{b}{b+2c}}+c\sqrt{\frac{c}{c+2a}}\ge\frac{a+b+c}{\sqrt{3}}\)