Áp dụng bđt Cô-si vào các số dương a,b,c:
\(\dfrac{a^2}{b}+b\ge2\sqrt{\dfrac{a^2}{b}\cdot b}=2\sqrt{a^2}=2a\Rightarrow\dfrac{a^2}{b}\ge2a-b\)
Chứng minh tương tự ta được:
\(\dfrac{b^2}{c}\ge2b-c;\dfrac{c^2}{a}\ge2c-a\)
\(\Rightarrow\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge2a+2b+2c-a-b-c=a+b+c\)
Dấu = xảy ra \(\Leftrightarrow a=b=c\)
Cho a,b,c>0. CMR
\(\dfrac{a^2}{b+c}+\dfrac{b^2}{a+c}+\dfrac{c^2}{c+a}\ge\dfrac{a+b+c}{2}\)
Cho a,b,c>0 thỏa mãn\(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}=1\). CMR
\(\dfrac{a^2}{a+b}+\dfrac{b^2}{b+c}+\dfrac{c^2}{a+b}\ge\dfrac{1}{2}\)
Cho a,b,c là số dương. CMR
\(\dfrac{a}{b^2}+\dfrac{b}{c^2}+\dfrac{c}{a^2}\ge\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
Cho a,b,c là các số dương thỏa abc=1.CmR:
\(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)
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 a,b,c là số dương thỏa mãn a+b+c=3. CMR
a/ \(\dfrac{a}{\sqrt{b+1}}+\dfrac{b}{\sqrt{c+1}}+\dfrac{c}{\sqrt{a+1}}\ge\dfrac{3\sqrt{2}}{2}\)
b/ \(\sqrt{\dfrac{a^3}{b+3}}+\sqrt{\dfrac{b^3}{c+3}}+\sqrt{\dfrac{c^3}{a+3}}\ge\dfrac{3}{2}\)
Cho a,b,c>0.Cmr
\(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\ge\dfrac{a+b}{b+c}+\dfrac{b+c}{a+b}+1\)
cho a,b,c\(\ge\)0,a+b+c=1.chứng minh rằng
\(\dfrac{a}{1+a^2}+\dfrac{b}{1+b^2}+\dfrac{c}{1+c^2}\le\dfrac{9}{10}\)
1. Tìm GTLN \(y=x^3\left(2-x\right)^5\)
2. Cho \(0\le a\le1\). Chứng minh rằng \(a\left(1-a^2\right)\)\(\le\dfrac{2}{3\sqrt{3}}\)
3. Cho a,b,c >0
CMR: \(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)
