\(VT=\frac{a}{\sqrt{a-1}}+\frac{b}{\sqrt{b-1}}+\frac{c}{\sqrt{c-1}}=\frac{a-1+1}{\sqrt{a-1}}+\frac{b-1+1}{\sqrt{b-1}}+\frac{c-1+1}{\sqrt{c-1}}\)
\(VT\ge\frac{2\sqrt{a-1}}{\sqrt{a-1}}+\frac{2\sqrt{b-1}}{\sqrt{b-1}}+\frac{2\sqrt{c-1}}{\sqrt{c-1}}=6\) (đpcm)
Dấu "=" xảy ra khi \(a=b=c=2\)
Cho a , b , c là các số thự dương thỏa mãn : \(\sqrt{a}+\sqrt{b}+\sqrt{c}=3\)
CMR \(a^2\sqrt{a}+b^2\sqrt{b}+c^2\sqrt{c}+\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}\ge6\)
Cho a,b,c>o tm abc=1
CMR\(\frac{b+c}{\sqrt{a}}+\frac{c+a}{\sqrt{b}}+\frac{a+b}{\sqrt{c}}\ge6\)
Cho \(a,b,c\ge0\)Chứng minh \(3\le\frac{\sqrt{a}+1}{\sqrt{b}+1}+\frac{\sqrt{b}+1}{\sqrt{c}+1}+\frac{\sqrt{c}+1}{\sqrt{a}+1}\le a+b+c+3\)
Cho a, b, c > 0 thoả mãn: a + b + c = \(\sqrt{a}+\sqrt{b}+\sqrt{c}\) = 2. Chứng minh: \(\frac{\sqrt{a}}{a+1}+\frac{\sqrt{b}}{b+1}+\frac{\sqrt{c}}{c+1}=\frac{2}{\sqrt{\left(1+a\right)\left(1+b\left(1+c\right)\right)}}\)
Cho a, b, c > 0 thoả mãn a + b + c = \(\sqrt{a}+\sqrt{b}+\sqrt{c}\) = 2. Chứng minh: \(\frac{\sqrt{a}}{a+1}+\frac{\sqrt{b}}{b+1}+\frac{\sqrt{c}}{c+1}=\frac{2}{\sqrt{\left(a+1\right)\left(b+1\right)\left(c+1\right)}}\)
Cho a,b,c >0 và a+b+c=\(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)Chứng minh
\(\sqrt{\frac{a^3}{1+3bc}}+\sqrt{\frac{b^3}{1+3ac}}+\sqrt{\frac{c^3}{1+3ba}}\ge\frac{3}{2}\)
Cho các số dương a,b,c thỏa mãn a+b+c=1.Chứng minh rằng
\(\sqrt{\frac{a}{1-a}}+\sqrt{\frac{b}{1-b}}+\sqrt{\frac{c}{1-c}}>2\)
Cho a,b,c>0 thỏa mãn \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1\). Chứng minh
\(\sqrt{a+bc}+\sqrt{b+ac}+\sqrt{c+ab}\ge\sqrt{abc}+\sqrt{a}+\sqrt{b}\)
Cho a, b, c dương. Chứng minh: \(\frac{1}{a\sqrt{3a+2b}}+\frac{1}{b\sqrt{3b+2c}}+\frac{1}{c\sqrt{3c+2a}}\ge\frac{3}{\sqrt{5abc}}\)
