\(\frac{a^2+b^2}{a-b}=\frac{a^2+b^2-2ab+2ab}{a-b}=\frac{\left(a-b\right)^2}{a-b}+\frac{2}{a-b}=a-b+\frac{2}{a-b}\ge2\sqrt{\frac{2\left(a-b\right)}{a-b}}=2\sqrt{2}\)
Dấu "=" xảy ra khi và chỉ khi \(\left\{{}\begin{matrix}ab=1\\a-b=\sqrt{2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=\frac{\sqrt{6}+\sqrt{2}}{2}\\b=\frac{\sqrt{6}-\sqrt{2}}{2}\end{matrix}\right.\)
Cho các số thực dương a, b, c. CMR:
\(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}+2\sqrt{\frac{a}{b+c}.\frac{b}{c+a}.\frac{c}{a+b}}\ge2\)
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, \(\left(\sqrt{a}+\sqrt{b}\right)^2\ge2\sqrt{2\left(a+b\right)\sqrt{ab}}\)
choa,b,c > 0. Cmr: \(\sqrt{\frac{2}{a}}+\sqrt{\frac{2}{b}}+\sqrt{\frac{2}{c}}\le\sqrt{\frac{a+b}{ab}}+\sqrt{\frac{b+c}{bc}}+\sqrt{\frac{c+a}{ca}}\)
Cho a,b,c>0 thoả mãn a2+b2+c2=1
CMR: \(\frac{a^2+ab+1}{\sqrt{a^2+3ab+c^2}}+\frac{b^2+bc+1}{\sqrt{b^2+3bc+a^2}}+\frac{c^2+ca+1}{\sqrt{c^2+3ac+b^2}}\ge\sqrt{5}\left(a+b+c\right)\)
cho a,b,c > 0 thỏa mãn abc =1. Cmr: \(\frac{1}{\sqrt{ab+a+2}}+\frac{1}{\sqrt{bc+b+2}}+\frac{1}{\sqrt{ca+c+2}}\le\frac{3}{2}\)
1.Cho a,b,c dương, a+b+c≤1.CMR: \(\frac{a^2+1}{a}+\frac{b^2+1}{b}+\frac{c^2+1}{c}\ge10\)
2.Cho a,b, c >0. CMR: \(\sqrt{x^2+\frac{1}{x^2}}+\sqrt{y^2+\frac{1}{y^2}}+\sqrt{z^2+\frac{1}{z^2}}\ge\sqrt{82};x+y+z\le1\)
Cho a;b;c>0.CMR:
\(\sqrt[3]{\frac{a^2+bc}{abc\left(b^2+c^2\right)}}+\sqrt[3]{\frac{b^2+ca}{abc\left(c^2+a^2\right)}}+\sqrt[3]{\frac{c^2+ab}{abc\left(a^2+b^2\right)}}\ge\frac{9}{a+b+c}\)
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 là các số thực dương t/m a + b+ c=abc. CMR :
\(\frac{1}{\sqrt{1+a^2}}+\frac{1}{\sqrt{1+b^2}}+\frac{1}{\sqrt{1+c^2}}\le\frac{3}{2}\)
