\(\Leftrightarrow ab^2+bc^2+ca^2\ge a^2b+b^2c+c^2a\)
\(\Leftrightarrow\left(c^2b-abc-b^2c+ab^2\right)+\left(ca^2+abc-ac^2-a^2b\right)\ge0\)
\(\Leftrightarrow b\left(c^2-ac-bc+ab\right)-a\left(c^2-ac-bc+ab\right)\ge0\)
\(\Leftrightarrow\left(b-a\right)\left(c^2-ac-bc+ab\right)\ge0\)
\(\Leftrightarrow\left(b-a\right)\left(c-b\right)\left(c-a\right)\ge0\) (luôn đúng do \(c\ge b\ge a>0\))
Cho 0<a≤b≤c .chứng minh rằng
\(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\)≥\(\dfrac{b}{a}+\dfrac{c}{b}+\dfrac{a}{c}\)
CMR với a, b, c > 0 thì :
a) \(\dfrac{bc}{a}+\dfrac{ac}{b}+\dfrac{ab}{c}\ge a+b+c\)
b)\(\dfrac{ab}{a+b}+\dfrac{bc}{b+c}+\dfrac{ca}{c+a}\le\dfrac{a+b+c}{2}\)
Cho a ≥ b ≥ c >0.
Chứng minh bất đẳng thức: \(\dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}\) ≤ \(\dfrac{b}{a}+\dfrac{a}{c}+\dfrac{c}{b}\)
Cho a, b, c > 0 .CMR: \(\dfrac{ab}{a+b}+\dfrac{bc}{b+c}+\dfrac{ac}{a+c}\) ≤ \(\dfrac{1}{2}\left(a+b+c\right)\)
Cho a, b, c > 0 .CMR: \(\dfrac{ab}{a+b}+\dfrac{bc}{b+c}+\dfrac{ac}{a+c}\) ≤ \(\dfrac{1}{2}\)
1,cho a,b,c. cm 1\(\ge\) a,b,c\(\ge\)0, a2 +b2+c2\(\le\)2
2, cho a,b,c>0.
CM: (a+b+c)(\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\ge9\))
3,cho a,b,c>0 CM:
\(\dfrac{a+b}{c}+\dfrac{b+c}{a}+\dfrac{c+a}{b}\ge6\)
mn giải giúp mk nha :)
Cho 3 số dương a, b, c. Chứng minh rằng:
\(\dfrac{1}{\dfrac{1}{a}+\dfrac{1}{b}}+\dfrac{1}{\dfrac{1}{b}+\dfrac{1}{c}}+\dfrac{1}{\dfrac{1}{c}+\dfrac{1}{a}}\le\dfrac{a+b+c}{2}\)
CMR với a, b, c > 0 thì
a) \(\dfrac{a^2}{b^2}+\dfrac{b^2}{a^2}\ge\dfrac{a}{b}+\dfrac{b}{a}\)
b) \(\dfrac{a^2}{b}+\dfrac{b^2}{c}+\dfrac{c^2}{a}\ge a+b+c\)
c) \(\dfrac{a^2}{b+c}+\dfrac{b^2}{c+a}+\dfrac{c^2}{a+b}\ge\dfrac{a+b+c}{2}\)
Cho 3 số thực a, b, c đôi một khác nhau thỏa mãn: \(\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}=0\)
CMR: \(\dfrac{a}{\left(b-c\right)^2}+\dfrac{b}{\left(c-a\right)^2}+\dfrac{c}{\left(a-b\right)^2}=0\)
