Theo BĐT AM - GM, ta có: \(\frac{a^3}{a^2+b^2}=a-\frac{ab^2}{a^2+b^2}\ge a-\frac{ab^2}{2ab}=a-\frac{b}{2}\)(1)
Tương tự ta có: \(\frac{b^3}{b^2+c^2}\ge b-\frac{c}{2}\)(2) ; \(\frac{c^3}{c^2+a^2}\ge c-\frac{a}{2}\)(3)
Cộng theo vế của 3 BĐT (1), (2), (3), ta được: \(\frac{a^3}{a^2+b^2}+\frac{b^3}{b^2+c^2}+\frac{c^3}{c^2+a^2}\ge\left(a+b+c\right)-\frac{a+b+c}{2}\)\(=\frac{a+b+c}{2}\)
Đẳng thức xảy ra khi a = b = c
Cho a,b,c > 0.Chứng minh rằng
a,\(\frac{1}{a}\)+\(\frac{1}{b}\)+\(\frac{1}{c}\)\(\ge\)\(\frac{2}{a+b}\)+\(\frac{2}{b+c}\)+\(\frac{2}{c+a}\)
b,\(\frac{4}{a}\)+\(\frac{5}{b}\)+\(\frac{3}{c}\)\(\ge\)\(4\left(\frac{3}{a+b}+\frac{2}{b+c}+\frac{1}{c+a}\right)\)
Cho a,b,c > 0.Chứng minh rằng
a,\(\frac{1}{a}\)+\(\frac{1}{b}\)+\(\frac{1}{c}\)\(\ge\)\(\frac{2}{a+b}\)+\(\frac{2}{b+c}\)+\(\frac{2}{c+a}\)
b,\(\frac{4}{a}\)+\(\frac{5}{b}\)+\(\frac{3}{c}\)\(\ge\)\(4\left(\frac{3}{a+b}+\frac{2}{b+c}+\frac{1}{c+a}\right)\)
cho a, b, c>0. CMR a\(\frac{a^3}{b}\ge a^2+ab-b^2\)
CM \(\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}\ge\frac{c}{b}+\frac{b}{a}+\frac{a}{c}\)
Cho a, b, c là độ dài 3 cạnh của tam giác CM \(\frac{1}{a+b-c}+\frac{1}{b+c-a}+\frac{1}{c+a-b}\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)
Cho a,b,c>0 CMR\(\frac{a^3}{b+c}+\frac{b^3}{c+a}+\frac{c^3}{a+b}\ge\frac{a^2+b^2+c^2}{2}\)
cho a,b,c>0. C/M a) \(\frac{_{-a+b+c}}{2a}+\frac{a-b+c}{2b}+\frac{a+b-c}{2c}\ge\frac{3}{2}\)\(\frac{3}{2}\)
b) \(\frac{a^2}{c}+\frac{b^2}{a}+\frac{c^2}{b}\ge a+b+c\)
Cho a,b,c dương. C/m\(\frac{a^3}{a^2+b^2}+\frac{b^3}{b^2+c^2}+\frac{c^3}{c^2+a^2}\ge\frac{a+b+c}{2}\)
Cho a,b,c dương và a+b+c=3. c/m\(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\ge\frac{3}{2}\)
và \(\frac{a}{a^2+1}+\frac{b}{b^2+1}+\frac{c}{c^2+1}\ge\frac{3}{2}\)
1. Cho \(a>0,b>0\). C/m \(\frac{a}{\sqrt{b}}-\sqrt{a}\ge\sqrt{b}-\frac{b}{\sqrt{a}}\)
2. Cho \(a\ne0,b\ne0\). C/m \(a^4+b^4\le\frac{a^6}{b^2}+\frac{b^6}{a^2}\)
3. C/m \(\frac{x^2}{y^2}+\frac{y^2}{z^2}+\frac{z^2}{x^2}\ge\frac{x}{y}+\frac{y}{z}+\frac{z}{x}\)
4. C/m \(\frac{x^2+y^2}{2}\ge\left(\frac{x+y}{2}\right)^2\)
5. \(\forall a,b>0\). C/m \(\frac{a^3}{b}+b^3>a^2+ab\)
Cho a,b,c >0 thỏa mãn a+b+c\(\le\)\(\frac{3}{2}\).Chứng minh
a,\(\frac{1}{a}\)+\(\frac{1}{b}\)+\(\frac{1}{c}\)\(\ge\)6
b,a+ b+ c+ \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\)\(\ge\)\(\frac{15}{2}\)
