\(\Leftrightarrow\left(a+b\right)\left(\frac{a+b}{2}+\frac{1}{4}\right)-\sqrt{ab}\left(\sqrt{a}+\sqrt{b}\right)\ge0\left(1\right)\)
Áp dụng BĐT Cô-si cho 2 số dương ta được:
\(VT\left(1\right)\ge\sqrt{ab}\left(a+b+\frac{1}{2}-\sqrt{a}-\sqrt{b}\right)\ge\sqrt{ab}\left[\left(\sqrt{a}-\frac{1}{2}\right)^2+\left(\sqrt{a}-\frac{1}{2}\right)^2\right]\ge0\)
\(\Rightarrow\left(1\right)\) Đúng.
\(\RightarrowĐpcm\)
cmr
\(\frac{\left(a+b\right)^2}{2}+\frac{a+b}{4}\ge a\sqrt{a}+b\sqrt{b}\)
Cho a,b,c >0 abc=1. CMR \(\frac{a^4}{b^2\left(c+a\right)}+\frac{b^4}{c^2\left(a+b\right)}+\frac{c^4}{a^2\left(b+c\right)}\ge\frac{a+b+c}{2}\)
Cho a,b\(\ge\)1 CMR:
a, \(a\sqrt{b-1}+b\sqrt{a-1}\le ab\)
b, \(a+\frac{1}{\left(a+1\right)^2}\ge\frac{3\sqrt[3]{4}}{4}\)
1. Choa>b>0 . CMR:
a. \(a+\frac{1}{b\left(a-b\right)}\ge3\)
b. \(a+\frac{4}{\left(a-b\right)\left(b+1\right)^2}\ge3\)
c. \(a+\frac{1}{b\left(a-b\right)^2}\ge2\sqrt{2}\)
Cho a,b,c là số dương. CMR:
1. \(\left(1+a\right)\left(1+b\right)\left(1+c\right)\ge\left(1+\sqrt[3]{abc}\right)^3\)
2. \(a^2\sqrt{bc}+b^2\sqrt{ac}+c^2\sqrt{ab}\le a^3+b^3+c^3\)
3. \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge\dfrac{a+b+c}{2}\)
cho các số thực dương a,b,c thỏa mãn \(\sqrt{a}+\sqrt{b}+\sqrt{c}=1\)
cmr \(\frac{a^2+bc}{\sqrt{2a^2\left(b+c\right)}}+\frac{b^2+ca}{\sqrt{2b^2\left(c+a\right)}}+\frac{c^2+ab}{\sqrt{2c^2\left(a+b\right)}}\ge1\)
Cho 3 số thực dương a,b,c. CMR \(\frac{a^2}{\sqrt{3a^2+8b^2+14ab}}+\frac{b^2}{\sqrt{3b^2+8c^2+14bc}}+\frac{c^2}{\sqrt{3c^2+8a^2+14ac}}\ge\frac{1}{5}\left(a+b+c\right)\)
Cho a,b,c > 0. Chứng minh:
\(\frac{a}{\sqrt[3]{4\left(b^3+c^3\right)}}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2}\)
Cho a>0, b>0 và c>0. Tính giá trị lớn nhất của biểu thức
A = \(\frac{a}{a+\sqrt[]{\left(a+b\right)\left(a+c\right)}}+\frac{b}{b+\sqrt[]{\left(b+c\right)\left(b+a\right)}}+\frac{c}{c+\sqrt[]{\left(c+a\right)\left(c+b\right)}}\)
