cho a,b,c > 0 thỏa mãn abc =1. Cmr: \(\frac{1}{\sqrt{ab+a+2}}+\frac{1}{\sqrt{bc+b+2}}+\frac{1}{\sqrt{ca+c+2}}\le\frac{3}{2}\)
cho a , b , c >0. Chứng minh các bất đẳng thức :
1, ab + bc + ca \(\ge\sqrt{abc}\left(\sqrt{a}+\sqrt{b}+\sqrt{c}\right)\)
2, \(\frac{ab}{c}+\frac{bc}{a}+\frac{ac}{b}\ge a+b+c\)
3, \(ab+\frac{a}{b}+\frac{b}{a}\ge a+b+1\)
4, \(\frac{a^3}{b}+\frac{b^3}{c}+\frac{c^3}{a}\ge ab+bc+ca\)
5, \(\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
cho a b c > 0. Chứng minh các bất đẳng thức :
1, \(\left(a+b+c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\ge9\)
2, \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}\ge\frac{16}{a+b+c+d}\)
3, ( 1+a+b) (a+b+ab) \(\ge9ab\)
4, \(\left(\sqrt{a}+\sqrt{b}\right)^8\ge64ab\left(a+b\right)^2\)
5, \(3a^3+7b^3\ge9ab^2\)
6, \(\left(\sqrt{a}+\sqrt{b}\right)^2\ge2\sqrt{2\left(a+b\right)\sqrt{ab}}\)
Cho a,b,c>0 thoả mãn a2+b2+c2=1
CMR: \(\frac{a^2+ab+1}{\sqrt{a^2+3ab+c^2}}+\frac{b^2+bc+1}{\sqrt{b^2+3bc+a^2}}+\frac{c^2+ca+1}{\sqrt{c^2+3ac+b^2}}\ge\sqrt{5}\left(a+b+c\right)\)
C/m các BĐT sau :
\(1.a^3-3a+4\ge b^3-3b
\)
\(2,\frac{1}{\frac{1}{a+c}+\frac{1}{b+d}}\ge\frac{1}{\frac{1}{a}+\frac{1}{b}}+\frac{1}{\frac{1}{c}+\frac{1}{d}}\) với a, b, c, d>0
\(3,a^3+b^3\ge\frac{1}{4};a+b\ge1\)
4, \(a^3+b^3\le a^4+b^4;a+b\ge2\)
5, \(\left(a+b\right)\left(a^3+b^3\right)\left(a^5+b^5\right)\le4\left(a^9+b^9\right);a,b\ge0\)
6, \(\frac{c+a}{\sqrt{a^2+c^2}}\ge\frac{c+b}{\sqrt{c^2+b^2}};a>b>0,c>\sqrt{ab}\)
Các bn làm đc bài nào thì giúp mk với, cảm ơn ạ !
1/ cho a,b,c >0
a+b+c=3:
chứng minh : \(\frac{a^2}{b+3c}+\frac{b^2}{c+3a}+\frac{c^2}{a+3b}\) ≥ \(\frac{3}{4}\)
2/a,b,c>0
a+b+c=6
chứng minh : S= \(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\) ≤ 6
Cho 3 số thực dương a,b,c thỏa mãn a+b+c=3.
CMR \(\frac{b+1}{8-\sqrt{a}}+\frac{c+1}{8-\sqrt{b}}+\frac{a+1}{8-\sqrt{c}}\le\frac{6}{7}\)
Cho a, b, c, d dương. CM:
1) \(\frac{a^2}{b^5}+\frac{b^2}{c^5}+\frac{c^2}{d^5}+\frac{d^2}{a^5}\ge\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\)
2) \(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge\frac{a+b+c}{\sqrt[3]{abc}}\)
3) \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{d^2}+\frac{d^2}{a^2}\ge\frac{a+b+c+d}{\sqrt[4]{abcd}}\)
4) \(\frac{1}{a^2+2bc}+\frac{1}{b^2+2ac}+\frac{1}{c^2+2ab}\ge9;a+b+c\le1\)
cho a,b,c > 0 thỏa mãn \(a+b+c\le\frac{3}{2}\)
Tìm GTNN của \(A=\sqrt{a^2+\frac{1}{b^2}}+\sqrt{b^2+\frac{1}{c^2}}+\sqrt{c^2+\frac{1}{a^2}}\)
