Cho a, b, c > 0 có ab + bc + ca = 1. Tìm GTNN \(P=\dfrac{a^3}{b^2+1}+\dfrac{b^3}{c^2+1}+\dfrac{c^3}{a^2+1}\)
1,Cho a,b,c>0 thỏa mãn a+b+c=abc.CMR:
\(\frac{bc}{a\left(1+bc\right)}+\frac{ca}{b\left(1+ca\right)}+\frac{ab}{c\left(1+ab\right)}\ge\frac{3\sqrt{3}}{4}\)
2,Cho a,b,c>0 thỏa mãn \(a^2+b^2+c^2=3\)
Tìm GTLN của P= \(\sqrt{\frac{a^2}{a^2+b+c}}+\sqrt{\frac{b^2}{b^2+c+a}}+\sqrt{\frac{c^2}{c^2+a+b}}\)
3,Cho a,b,c>0 thỏa mãn a+b+c=3.
Tìm GTLN của Q= \(2\sqrt{abc}\left(\frac{1}{\sqrt{3a^2+4b^2+5}}+\frac{1}{\sqrt{3b^2+4c^2+5}}+\frac{1}{\sqrt{3c^2+4a^2+5}}\right)\)
4,Cho a,b,c>0.
Tìm GTLN của P= \(\frac{\sqrt{ab}}{c+3\sqrt{ab}}+\frac{\sqrt{bc}}{a+3\sqrt{bc}}+\frac{\sqrt{ca}}{b+3\sqrt{ca}}\)
Cho a, b, c > 0 và \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=\dfrac{1}{3}\) .
Tìm MAX : A= \(\dfrac{1}{a^2+bc}+\dfrac{1}{b^2+ca}+\dfrac{1}{c^2+ab}\)
Cho a,b,c>0 và \(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}=\dfrac{1}{3}\). Tìm GTLN P=\(\dfrac{1}{a^2+bc}+\dfrac{1}{b^2+ca}+\dfrac{1}{c^2+ab}\)
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 dương, 0<a,b,c<1/2, thỏa a+b+c=1. CMR: a^3+b^3+c^3+2*(ab+bc+ca)<=25/32
Cho a, b, c > 0 có ab + bc + ca = 1. Tìm GTNN \(P=\frac{a^3}{b^2+1}+\frac{b^3}{c^2+1}+\frac{c^3}{a^2+1}\)
cho a,b,c>0 thỏa mãn \(a^2+b^2+c^2=1\).CMR
\(\dfrac{\sqrt{ab+2c^2}}{\sqrt{1+ab-c^2}}+\dfrac{\sqrt{bc+2a^2}}{\sqrt{1+bc-a^2}}+\dfrac{\sqrt{ca+2b^2}}{\sqrt{1+ca-b^2}}\ge2+ab+bc+ca\)
cho a,b,c >0 va abc=1 c/m
\(\frac{1+ab^2}{c^3}+\frac{1+bc^2}{a^3}+\frac{1+ca^2}{b^3}>=\frac{18}{a^3+b^3+c^3}\)
