Lời giải:
Áp dụng BĐT Cô-si cho 3 dương:
\(a+b+c\geq 3\sqrt[3]{abc}\)
\(a^2+b^2+c^2\geq 3\sqrt[3]{a^2b^2c^2}\)
Nhân theo vế thu được:
\((a+b+c)(a^2+b^2+c^2)\geq 3\sqrt[3]{abc}.3\sqrt[3]{a^2b^2c^2}=9abc\)
Ta có đpcm.
Dấu "=" xảy ra khi $a=b=c$
Cho a,b,c > 0. Cmr: \(\frac{a\left(b+c\right)}{a^2+\left(b+c\right)^2}+\frac{b\left(c+a\right)}{b^2+\left(c+a\right)^2}+\frac{c\left(a+b\right)}{c^2+\left(a+b\right)^2}\le\frac{6}{5}\)
Cho a,b,c>0. CMR:
\(\frac{\left(a+b\right)^2}{c}+\frac{\left(b+c\right)^2}{a}+\frac{\left(c+a\right)^2}{b}\ge4\left(a+b+c\right)\)
1. cho \(0< a\le b\le c\) . Cmr: \(\frac{2a^2}{b^2+c^2}+\frac{2b^2}{c^2+a^2}+\frac{2c^2}{a^2+b^2}\le\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{a}\)
2. cho \(a,b,c\ge0\). cmr: \(a^2+b^2+c^2+3\sqrt[3]{\left(abc\right)^2}\ge2\left(ab+bc+ca\right)\)
3. \(a,b,c>0.\) Cmr: \(\sqrt{\left(a^2b+b^2c+c^2a\right)\left(ab^2+bc^2+ca^2\right)}\ge abc+\sqrt[3]{\left(a^3+abc\right)\left(b^3+abc\right)\left(c^3+abc\right)}\)
4. \(a,b,c>0\). Tìm Min \(P=\left(\frac{a}{a+b}\right)^4+\left(\frac{b}{b+c}\right)^4+\left(\frac{c}{c+a}\right)^4\)
Cho a,b,c>0 thỏa mãn a+b+c=3 CMR:
\(\dfrac{a^4}{\left(a+2\right)\left(b+2\right)}+\dfrac{b^4}{\left(b+2\right)\left(c+2\right)}+\dfrac{c^4}{\left(c+2\right)\left(a+2\right)}\ge\dfrac{1}{3}\)
1. Cho a,b,c>0. Cmr: \(\frac{a}{\left(b+c\right)^2}+\frac{b}{\left(c+a\right)^2}+\frac{c}{\left(a+b\right)^2}\ge\frac{9}{4\left(a+b+c\right)}\)
Cho \(\dfrac{a}{b-c}+\dfrac{b}{c-a}+\dfrac{c}{a-b}=0\) . CMR:
\(\dfrac{a}{\left(b-c\right)^2}+\dfrac{b}{\left(c-a\right)^2}+\dfrac{c}{\left(a-b\right)^2}=0\)
Cho a, b, c>0; abc=1. Cmr:
\(\dfrac{a^3}{b\left(c+2\right)}+\dfrac{b^3}{c\left(a+2\right)}+\dfrac{c^3}{a\left(b+2\right)}\ge1\)
Sao em làm chỉ ra >=3 thôi ạ)):
Cho a,b,c \(\ge0\)
Cmr: \(a^2\left(b+c-a\right)+b^2\left(c+a-b\right)+c^2\left(a+b-c\right)\le3abc\)
Cho a,b,c\(\ge0\).CMR:\(\left(a^2+b^2+c^2\right)^2\ge4\left(a+b+c\right)\left(a-b\right)\left(b-c\right)\left(c-a\right)\)
