Chứng minh :
a, \(\dfrac{a+b+c}{3}\dfrac{>}{ }\sqrt{\dfrac{ab+bc+ca}{3}}\) với a,b,c>0
b,\(\dfrac{a^2+b^2+c^2}{3}\dfrac{>}{ }\left(\dfrac{a+b+c}{3}\right)^2\)
c,\(\dfrac{x^2+2}{\sqrt{x^2+1}}\dfrac{>}{ }2\)
d,\(\dfrac{a^3+b^3}{2}\dfrac{>}{ }\left(\dfrac{a+b}{2}\right)^3\)
Cho a,b,c dương. Chứng minh
\(\dfrac{1}{\left(a+b\right)^2}+\dfrac{1}{\left(b+c\right)^2}+\dfrac{1}{\left(c+a\right)^2}\ge\dfrac{3\sqrt{3abc\left(a+b+c\right)}.\left(a+b+c\right)^2}{4\left(ab+bc+ca\right)^3}\)
Cho a,b,c > 0. Chứng minh: \(\dfrac{19b^3-a^3}{ab+5b^2}+\dfrac{19c^3-b^3}{bc+5c^2}+\dfrac{19a^3-c^3}{ac+5a^2}\le3\left(a+b+c\right)\)
giúp mình vài câu sau nha
thanks nhiều
1. cho a,b,c > 0
C/m: \(\dfrac{a^3+b^3}{ab}+\dfrac{b^3+c^3}{bc}+\dfrac{c^3+a^3}{ca}>=2\left(a+b+c\right)\)
2. cho a,b,c > 0
C/m: \(\dfrac{a^4}{bc^2}+\dfrac{b^4}{ca^2}+\dfrac{c^4}{ab^2}>=a+b+c\)
3. cho a,b,c > 0 và \(a^2+b^2+c^2=3\)
C/m: \(\dfrac{a^3}{b+c}+\dfrac{b^3}{c+a}+\dfrac{c^3}{a+b}>=\dfrac{3}{2}\)
4. cho a,b,c > 0
C/m: \(\dfrac{a^3}{a^2+b^2}+\dfrac{b^3}{b^2+c^2}+\dfrac{c^3}{c^2+a^2}>=\dfrac{a+b+c}{2}\)
Cho a , b , c > 0 thỏa mãn \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=1\)
Chứng minh rằng \(\dfrac{a^2}{a+bc}+\dfrac{b^2}{b+ca}+\dfrac{c^2}{c+ab}\ge\dfrac{a+b+c}{4}\)
Cho a , b , c > 0 thỏa mãn \(a+b+c=3\)
Chứng minh rằng \(\dfrac{ab}{\sqrt{c^2+3}}+\dfrac{bc}{\sqrt{a^2+3}}+\dfrac{ca}{\sqrt{b^2+3}}\le\dfrac{a}{b+c}+\dfrac{b}{c+a}+\dfrac{c}{a+b}\)
Cho a,b,c >0 thỏa a+b+c=3.Chứng minh rằng
\(\dfrac{a}{ab+1}+\dfrac{b}{bc+1}+\dfrac{c}{ca+1}\ge\dfrac{3}{2}\)
Cho a,b,c>0.Chứng minh rằng:\(\dfrac{a^2+b^2}{a+b}+\dfrac{b^2+c^2}{b+c}+\dfrac{c^2+a^2}{a+c}\le\dfrac{3.\left(a^2+b^2+c^2\right)}{a+b+c}\)
Chứng minh bất đẳng thức : \(\dfrac{a+b}{a^2+b^2}+\dfrac{b+c}{b^2+c^2}+\dfrac{c+a}{c^2+a^2}\)\(\forall a,b,c>0;a+b+c=ab+ac+ca\)
