\(\sqrt{\frac{bc}{a\left(3b+a\right)}}+\sqrt{\frac{ac}{b\left(3c+b\right)}}+\sqrt{\frac{ab}{c\left(3a+c\right)}}\ge\frac{3}{2}\)
Cho a,b,c là các số thực dương. Chứng minh rằng:
\(\dfrac{3a^3+7b^3}{2a+3b}+\dfrac{3b^3+7c^3}{2b+3c}+\dfrac{3c^3+7a^3}{2c+3a}\ge3\left(a^2+b^2+c^2\right)-\left(ab+bc+ca\right)\)
Cho a, b, c dương và \(ab^2+bc^2+ca^2=3\). CM:
\(\dfrac{2a^5+3b^5}{ab}+\dfrac{2b^5+3c^5}{bc}+\dfrac{2c^5+3a^5}{ac}\ge15\left(a^2+b^2+c^2-2\right)\)
1.\(\left\{{}\begin{matrix}a,b,c>0\\ab+bc+ca=3\end{matrix}\right.\) Cmr: \(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge\frac{3}{2}\)
2.\(a,b,c>0\). Cmr: \(\frac{ab^2}{a^2+2b^2+c^2}+\frac{bc^2}{b^2+2c^2+a^2}+\frac{ca^2}{c^2+2a^2+b^2}\le\frac{a+b+c}{4}\)
3. \(a,b,c>0\). Cmr: \(\frac{ab}{a+3b+2c}+\frac{bc}{b+3c+2a}+\frac{ca}{c+3a+2b}\le\frac{a+b+c}{6}\)
Cho a,b,c là các số dương thỏa mãn điều kiên a+b+c=3. Tìm GTLN của biểu thức:
P=\(\frac{5b^3-a^3}{ab+3b^2}+\frac{5c^3-b^3}{bc+3c^2}+\frac{5a^3-c^3}{ca+3a^2}\)
Cho ba số thực dương thỏa mãn a+b+c=1 . Tìm GTLN của biểu thức:
\(P=\dfrac{a}{9a^3+3b^2+c}+\dfrac{b}{9b^3+3c^2+a}+\dfrac{c}{9c^3+3a^2+b}+2018\left(ab+bc+ca\right)\)
1. a) cho \(1\le a,b,c\le2\). Tìm max \(P=\left(x+y\right)\left(\frac{1}{x}+\frac{1}{y}\right)\)
b) \(\left\{{}\begin{matrix}a,b,c\ge0\\a+b+c=1\end{matrix}\right.\). Cmr: \(\sqrt{\frac{3a^2+1}{3b^2+1}}+\sqrt{\frac{3b^2+1}{3c^2+1}}+\sqrt{\frac{3c^2+1}{3a^2+1}}\le\frac{7}{2}\)
2.a) \(a,b\ge0;c\ge1;a+b+c=2\). cmr: \(\left(6-a^2-b^2-c^2\right)\left(2-abc\right)\le8\)
b) \(\left\{{}\begin{matrix}a+b\le2\\a^2+b^2+ab=3\end{matrix}\right.\). Tìm max,min \(P=a^2+b^2-ab\)
a,b,c>0, biết a+b+c=3
CMR a)\(\frac{ab}{\sqrt{a^2+3b^2}}+\frac{bc}{\sqrt{b^2+3c^2}}+\frac{ac}{\sqrt{c^2+3a^2}}\)≤\(\frac{3}{2}\)
b)\(\frac{a}{\sqrt{b^2+3}}+\frac{b}{\sqrt{c^2+3}}+\frac{c}{\sqrt{a^2+3}}\)≥\(\frac{3}{2}\)
Cho a, b, c dương thỏa abc = 1. Chứng minh: \(\frac{1}{a^3\left(7b+3c\right)}+\frac{1}{b^3\left(7c+3a\right)}+\frac{1}{c^3\left(7a+3b\right)}\ge\frac{1}{10}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)