Question

[Trần Phúc]

Cmr:

\(\forall a,b,c\ge0\)

\(\frac{a^3b}{c}+\frac{a^3c}{b}+\frac{b^3c}{a}+\frac{b^3a}{c}+\frac{c^3a}{b}+\frac{c^3b}{a}\ge6abc\)

Ai nhanh mình k cho nhé! 

Answer 1

[Witch Rose]

Ad BĐT Cauchy cho 6 số: 

\(\frac{a^3b}{c}+\frac{a^3c}{b}+\frac{b^3c}{a}+\frac{b^3a}{c}+\frac{c^3a}{b}+\frac{c^3b}{a}\ge6\sqrt[6]{\frac{a^8b^8c^8}{a^2b^2c^2}}=6abc\)

Dấu = xr khi a=b=c

Answer 2

[l҉o҉n҉g҉ d҉z҉]

Áp dụng bất đẳng thức Cauchy cho VT ta được :

\(VT\ge6\sqrt[6]{\frac{a^3b}{c}\cdot\frac{a^3c}{b}\cdot\frac{b^3c}{a}\cdot\frac{b^3a}{c}\cdot\frac{c^3a}{b}\cdot\frac{c^3b}{a}}=6\sqrt[6]{\frac{a^8b^8c^8}{a^2b^2c^2}}=6\sqrt[6]{a^6b^6c^6}=6abc=VP\)

=> đpcm

Dấu "=" xảy ra <=> a = b = c

\(\)