Cho a, b, c là các số thực dương. Chứng minh rằng:
\(\sqrt{\frac{bc}{a\left(3b+a\right)}}+\sqrt{\frac{ca}{b\left(3c+b\right)}}+\sqrt{\frac{ab}{c\left(3a+c\right)}}\ge\frac{3}{2}\)
Bài 1 :Cho a,b,c dương thỏa mãn a+b+c=2
CMR \(\frac{bc}{\sqrt{3a^2+4}}+\frac{ca}{\sqrt{3b^2+4}}+\frac{ab}{\sqrt{3c^2+4}}\ge\frac{\sqrt{3}}{3}\)
Bài 2:Cho a,b,c>0. CMR
\(\left(a+b\right)\left(b+c\right)\left(c+a\right)\ge\frac{8}{9}\left(a+b+c\right)\left(ab+bc+ca\right)\)
\(\frac{a+b+c}{\sqrt{a\left(a+3b\right)}+\sqrt{b\left(b+3c\right)}+\sqrt{c\left(c+3a\right)}}\ge\) \(\frac{1}{2}\) vòi a,b,c là số dương? giúp mình với ah
- gợi ý: ADBDT: \(\frac{x+y}{2}\ge\sqrt{xy}\)
1,Cho a,b,c>0 thỏa mãn a+b+c=abc.CMR:
\(\frac{bc}{a\left(1+bc\right)}+\frac{ca}{b\left(1+ca\right)}+\frac{ab}{c\left(1+ab\right)}\ge\frac{3\sqrt{3}}{4}\)
2,Cho a,b,c>0 thỏa mãn \(a^2+b^2+c^2=3\)
Tìm GTLN của P= \(\sqrt{\frac{a^2}{a^2+b+c}}+\sqrt{\frac{b^2}{b^2+c+a}}+\sqrt{\frac{c^2}{c^2+a+b}}\)
3,Cho a,b,c>0 thỏa mãn a+b+c=3.
Tìm GTLN của Q= \(2\sqrt{abc}\left(\frac{1}{\sqrt{3a^2+4b^2+5}}+\frac{1}{\sqrt{3b^2+4c^2+5}}+\frac{1}{\sqrt{3c^2+4a^2+5}}\right)\)
4,Cho a,b,c>0.
Tìm GTLN của P= \(\frac{\sqrt{ab}}{c+3\sqrt{ab}}+\frac{\sqrt{bc}}{a+3\sqrt{bc}}+\frac{\sqrt{ca}}{b+3\sqrt{ca}}\)
Cho các số thực dương a,b,c thỏa mãn \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\ge1\). CMR:
\(\frac{a+b}{\sqrt{ab+c}}+\frac{b+c}{\sqrt{bc+a}}+\frac{c+a}{\sqrt{ca+b}}\ge3\sqrt[6]{abc}\)
Giải:
\(GT\Leftrightarrow ab+bc+ca\ge abc\)
\(\Rightarrow ab\le\frac{ab+bc+ca}{c}\)
\(\Rightarrow\frac{a+b}{\sqrt{ab+c}}\ge\frac{a+b}{\sqrt{\frac{ab+bc+ca}{c}+c}}=\frac{\left(a+b\right)\sqrt{c}}{\sqrt{\left(c+a\right)\left(c+b\right)}}\)
Tương tự rồi cộng lại: \(VT\ge\frac{\left(a+b\right)\sqrt{c}}{\sqrt{\left(c+a\right)\left(c+b\right)}}+\frac{\left(b+c\right)\sqrt{a}}{\sqrt{\left(a+b\right)\left(a+c\right)}}+\frac{\left(c+a\right)\sqrt{c}}{\sqrt{\left(b+a\right)\left(b+c\right)}}\)\(\ge3\sqrt[3]{\sqrt{abc}}=3\sqrt[6]{abc}\)
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Bài 1: Cho a,b>0. Chứng minh \(\sqrt[3]{\left(a+b\right)\left(\frac{1}{a}+\frac{1}{b}\right)}< \sqrt[3]{\frac{a}{b}}+\sqrt[3]{\frac{b}{a}}\)
Bài 2: Cho a,b>0. Chứng minh \(\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\ge\frac{2\sqrt{2}}{\sqrt{a+b}}\)
Bài 3: Cho a,b,c>0. Chứng minh \(\frac{5b^3-a^3}{ab+3b^2}+\frac{5c^3-b^3}{bc+3c^2}+\frac{5a^3-c^3}{ca+3a^2}\le a+b+c\)
\(\sqrt{ab}+\sqrt{bc}+\sqrt{ac}\ge3\sqrt[6]{abc}=3\)
Ta có \(\frac{a}{a+2}+\frac{b}{b+2}+\frac{c}{c+2}\ge\frac{\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)^2}{a+b+c+6}=\frac{a+b+c+2\left(\sqrt{ab}+\sqrt{bc}+\sqrt{ac}\right)}{a+b+c+6}\ge1\)
=> \(\frac{a}{a+2}+\frac{b}{b+2}+\frac{c}{c+2}\ge1\)
=> \(\left(\frac{1}{2}-\frac{1}{a+2}\right)+\left(\frac{1}{2}-\frac{1}{b+1}\right)+\left(\frac{1}{2}-\frac{1}{c+1}\right)\ge\frac{1}{2}\)
=> \(\frac{1}{a+2}+\frac{1}{b+2}+\frac{1}{c+2}\le1\)(ĐPCM)
Cho a, b, c là các số dương . CMR:
\(\frac{a\left(b+2c\right)}{\sqrt{3b^2+6c^2}}+\frac{b\left(c+2a\right)}{\sqrt{3c^2+6a^2}}+\frac{c\left(a+2b\right)}{\sqrt{3a^2+6b^2}}\le a+b+c\)
Cho a,b > 0 . CMR : \(\frac{a+b}{\sqrt{a\left(3a+b\right)}+\sqrt{b\left(3b+a\right)}}\ge\frac{1}{2}\left(1\right)\)