a, Ta có : \(a^2+b^2\ge2ab\) ( cauchuy )
\(\Rightarrow a^2+2ab+b^2=\left(a+b\right)^2\ge4ab\)
\(\Rightarrow\dfrac{a+b}{ab}=\dfrac{a}{ab}+\dfrac{b}{ab}=\dfrac{1}{a}+\dfrac{1}{b}\ge\dfrac{4}{a+b}\)
b, Ta có : \(a^2+b^2\ge2ab\) ( cauchuy )
\(\Rightarrow ab\le\dfrac{a^2+b^2}{2}\)
cho a,b,c>0.CMR
\(\dfrac{a+b}{ab+c^2}+\dfrac{b+c}{bc+a^2}+\dfrac{c+a}{ca+b^2}\le\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\)
cho 0<a,b,c<2 cmr \(\dfrac{1}{2-a}\)+\(\dfrac{1}{2-b}\)+\(\dfrac{1}{2-c}\)>=\(\dfrac{a^2+b^2+c^2+ab+bc+ac}{2}\)
Cho a, b, c > 0 thỏa mãn \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\). CMR:
\(\dfrac{a^2}{a+bc}+\dfrac{b^2}{b+ac}+\dfrac{c^2}{c+ab}\ge\dfrac{a+b+c}{4}\)
\(Cho a,b,c>0. Cmr: \dfrac{a^3b}{1+ab^2}+\dfrac{b^3c}{1+bc^2}+\dfrac{c^3a}{1+ca^2}>\dfrac{abc(a+b+c)}{1+abc}\)
a,b,c>0 thỏa mãn `a^4 +b^4 +c^4 =3`. CMR \(\dfrac{a^2}{b^3+1}+\dfrac{b^2}{c^3+1}+\dfrac{c^2}{a^3+1}>=\dfrac{3}{2}\)
a,b,c>0 thỏa mãn `a^4 +b^4 +c^4 =3`. CMR: \(\dfrac{a^2}{b^3+1}+\dfrac{b^2}{c^3+1}+\dfrac{c^2}{a^3+1}>=\dfrac{3}{2}\)
Cho a, b, c > 0 thỏa mãn abc = 8. CMR:
\(\dfrac{a}{ac+4}+\dfrac{b}{ab+4}+\dfrac{c}{bc+4}\le\dfrac{a^2+b^2+c^2}{16}\)
Cho a,b,c>0 thỏa mãn ab+bc+ca=1. CMR:
\(\left(\dfrac{a}{\sqrt{a^2+1}}+\dfrac{b}{\sqrt{b^2+1}}+\dfrac{c}{\sqrt{c^2+1}}\right)^3\le\dfrac{3}{2}\left(\dfrac{1}{a^2+1}+\dfrac{1}{b^2+1}+\dfrac{1}{c^2+1}\right)\)
cho a,b,c>0 thỏa mãn abc=1.
CMR:\(\dfrac{a}{ab+1}+\dfrac{b}{bc+1}+\dfrac{c}{ca+1}\ge\dfrac{3}{2}\)
