Các câu hỏi tương tự
\(\frac{b^2c^3}{a^2+\left(b+c\right)^3}+\frac{c^2a^3}{b^2+\left(c+a\right)^3}+\frac{a^2b^3}{c^2+\left(a+b\right)^3}\ge\frac{9abc}{4\left(3abc+a^2c+b^2a+c^2b\right)}\)voi a,b,c>0
Cho các số thực a, b, c > 0. Chứng minh rằng :
\(\frac{a^2}{\left(2a+b\right)\left(2a+c\right)}+\frac{b^2}{\left(2b+a\right)\left(2b+c\right)}+\frac{c^2}{\left(2c+a\right)\left(2c+b\right)}\ge\frac{1}{3}\)
Cho a,b,c>0.Tìm Min A=\(\frac{a^4}{b^3\left(c+2a\right)}+\frac{b^4}{c^3\left(a+2b\right)}+\frac{c^4}{a^3\left(b+2c\right)}\)
Cho a,b,c là các số dương . Chứng minh rằng:
a) \(A=\frac{a}{2a+b+c}+\frac{b}{2b+c+a}+\frac{c}{2c+a+b}\le\frac{3}{4}\)
b) \(B=\left(a^5-a^2+3\right)\left(b^5-b^2+3\right)\left(c^5-c^2+3\right)\ge\left(a+b+c\right)^3\)
1/Cho các số thực dương chứng minh:frac{3left(a^4+b^4+c^4right)}{left(a^2+b^2+c^2right)^2}+frac{ab+bc+ca}{a^2+b^2+c^2}ge22/Cho a,b dương.Chứng minh:left(frac{a}{b}+frac{b}{a}right)+4sqrt{2}frac{a+b}{sqrt{a^2+b^2}}ge103/ Cho các số thực dương. Chứng minh: left(a^2+2bcright)left(b^2+2caright)left(c^2+2abright)ge abcleft(a+2bright)left(b+2cright)left(c+2aright)
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1/Cho các số thực dương chứng minh:\(\frac{3\left(a^4+b^4+c^4\right)}{\left(a^2+b^2+c^2\right)^2}+\frac{ab+bc+ca}{a^2+b^2+c^2}\ge2\)
2/Cho a,b dương.Chứng minh:\(\left(\frac{a}{b}+\frac{b}{a}\right)+4\sqrt{2}\frac{a+b}{\sqrt{a^2+b^2}}\ge10\)
3/ Cho các số thực dương. Chứng minh: \(\left(a^2+2bc\right)\left(b^2+2ca\right)\left(c^2+2ab\right)\ge abc\left(a+2b\right)\left(b+2c\right)\left(c+2a\right)\)
Pfrac{a}{sqrt{left(b+1right)left(b^2-b+1right)}}+frac{b}{sqrt{left(c+1right)left(c^2-c+1right)}}+frac{c}{sqrt{left(a+1right)left(a^2-a+1right)}}gefrac{2a}{b^2+2}+frac{2b}{c^2+2}+frac{2c}{a^2+2}left(a+b+cright)-left(frac{ab^2}{b^2+2}+frac{bc^2}{c^2+2}+frac{ca^2}{a^2+2}right)6-left(frac{2ab^2}{b^2+4+b^2}+frac{2bc^2}{c^2+4+c^2}+frac{2ca^2}{a^2+4+a^2}right)ge6-left(frac{2ab}{b+4}+frac{2bc}{c+4}+frac{2ca}{a+4}right)6-left(2a+2b+2c-frac{8a}{b+4}-frac{8b}{c+4}-frac{8c}{a+4}right)frac{8a}{b+4}+frac{8b}{...
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\(P=\frac{a}{\sqrt{\left(b+1\right)\left(b^2-b+1\right)}}+\frac{b}{\sqrt{\left(c+1\right)\left(c^2-c+1\right)}}+\frac{c}{\sqrt{\left(a+1\right)\left(a^2-a+1\right)}}\)
\(\ge\frac{2a}{b^2+2}+\frac{2b}{c^2+2}+\frac{2c}{a^2+2}=\left(a+b+c\right)-\left(\frac{ab^2}{b^2+2}+\frac{bc^2}{c^2+2}+\frac{ca^2}{a^2+2}\right)\)
\(=6-\left(\frac{2ab^2}{b^2+4+b^2}+\frac{2bc^2}{c^2+4+c^2}+\frac{2ca^2}{a^2+4+a^2}\right)\ge6-\left(\frac{2ab}{b+4}+\frac{2bc}{c+4}+\frac{2ca}{a+4}\right)\)
\(=6-\left(2a+2b+2c-\frac{8a}{b+4}-\frac{8b}{c+4}-\frac{8c}{a+4}\right)\)
\(=\frac{8a}{b+4}+\frac{8b}{c+4}+\frac{8c}{a+4}-6=\frac{8a^2}{ab+4a}+\frac{8b^2}{bc+4b}+\frac{8c^2}{ca+4c}-6\)
\(\ge\frac{8\left(a+b+c\right)^2}{\left(ab+bc+ca\right)+4\left(a+b+c\right)}-6\ge\frac{288}{\frac{\left(a+b+c\right)^2}{3}+24}-6=2\)
20 tháng 2 2020 lúc 14:50
Cho a,b,c > 0 thỏa mãn a + b + c = 3.
Chứng minh rằng: \(\frac{a^4}{\left(b+c\right)\left(b^2+c^2\right)}+\frac{b^4}{\left(c+a\right)\left(c^2+a^2\right)}+\frac{c^4}{\left(a+b\right)\left(a^2+b^2\right)}\ge\frac{3}{4}\)
Cho a,b,c>0 thỏa mãn \(a+b+c\le3\)
Chứng minh \(\frac{1}{\left(2a+b\right)\left(2c+b\right)}+\frac{1}{\left(2b+c\right)\left(2a+c\right)}+\frac{1}{\left(2c+a\right)\left(2b+a\right)}\ge\frac{3}{\left(a+b+c\right)^2}\)
cho các số thực a,b không âm:
Chứng minh rằng: \(\left(a^2+b+\frac{3}{4}\right)+\left(b^2+a+\frac{3}{4}\right)\ge\left(2a+\frac{1}{2}\right)\left(2b+\frac{1}{2}\right)\)