\(a^2+b^2+c^2\ge a\left(b+c\right)\Leftrightarrow2a^2+2b^2+2c^2-2ab-2ac\ge0\Leftrightarrow\left(a-b\right)^2+\left(a-c\right)^2+b^2+c^2\ge0\) Biểu thức luôn đúng với mọi a , b , c .
Dấu ''='' xảy ra khi a = b = c = 0
1. Cho a,b,c là các số thực dương thỏa a+b+c=3. Cmr \(\dfrac{a^2}{a+2b^2}+\dfrac{b^2}{b+2c^2}+\dfrac{c^2}{c+2a^2}\ge1\)
2. Cho a,b,c là các số thực dương thỏa \(a^2+b^2+c^2=1\). Cmr: \(\dfrac{a}{1+b^2}+\dfrac{b}{1+c^2}+\dfrac{c}{1+a^2}\ge\dfrac{3}{4}\left(a\sqrt{a}+b\sqrt{b}+c\sqrt{c}\right)^2\)
3.Cho a,b,c là các số thực dương thỏa \(a^2+b^2+c^2=3\). Cmr:\(\sqrt{\dfrac{a^2}{b+b^2+c}}+\sqrt{\dfrac{b^2}{c+c^2+a}}+\sqrt{\dfrac{c^2}{a+a^2+b}}\le3\)
Bài 1:Cho 0<=a;b;c<=2.a+b+c=3
CM:3<=a^3+b^3+c^3-3(a-1)(b-1)(c-1)<=9
Bài 2: Cho -1<=a;b;c<=2.a+b+c=0.CM:
a,a^2+b^2+c^2<=6
b,2abc<=a^2+b^2+c^2<=2abc+2
c,a^2+b^2+c^2<=8-abc
1)cho a,b,c >0. \(cmr:\dfrac{1}{a^2+bc}+\dfrac{1}{b^2+ca}+\dfrac{1}{c^2+ab}\le\dfrac{a+b+c}{2abc}\)
2) cho a,b,c>0 và a+b+c=1. \(cmr:\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\left(1+\dfrac{1}{c}\right)\ge64\)
3) cho a,b,c>0. \(cme:\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)
4) cho a,b,c>0 .\(cmr:\dfrac{a^3}{b^3}+\dfrac{b^3}{c^3}+\dfrac{c^3}{a^3}\ge\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\)
5)cho a,b,c>0. cmr: \(\dfrac{1}{a\left(a+b\right)}+\dfrac{1}{b\left(b+c\right)}+\dfrac{1}{c\left(c+a\right)}\ge\dfrac{27}{2\left(a+b+c\right)^2}\)
Cho a, b, c > 0. Chứng minh \(\dfrac{a^2}{b^2}+\dfrac{b^2}{c^2}+\dfrac{c^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)
chứng minh: a) \(\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge\frac{3}{2},vớia,b,c>0\)
b) \(\frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}\ge\frac{a+b+c}{2}\)
Chứng minh rằng : Với 3 số dương ta có:
(a^2/b + b^2/c + c^2/a) +( a+b+c) >= [6(a^2 +b^2 + c^2)]/(a+b+c)
cho a,b,c là số thực dương. Cmr:
1.\(\dfrac{a}{b^2+bc+c^2}+\dfrac{b}{c^2+ca+a^2}+\dfrac{c}{a^2+ab+b^2}\ge\dfrac{a+b+c}{ab+bc+ca}\)
2.\(\left(a+b+c\right)\left(\dfrac{a}{\left(b+c\right)^2}+\dfrac{b}{\left(c+a\right)^2}+\dfrac{c}{\left(a+b\right)^2}\right)\ge\dfrac{9}{4}\)
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 \(a^2+b^2+c^2=3\) cmr
\(\sqrt{\dfrac{a^2}{b+b^2+c}}+\sqrt{\dfrac{b^2}{c+c^2+a}}+\sqrt{\dfrac{c^2}{a+a^2+b}}\le\sqrt{3}\)
Cho a,b,c > 0. CMR:
\(\dfrac{a^3}{a^2+b^2}+\dfrac{b^3}{b^2+c^2}+\dfrac{c^3}{c^2+a^2}\ge\dfrac{a+b+c}{2}\)
