Question

Cho x, y, z dương thoả mãn

\(\frac{1}{1+x}+\frac{1}{1+y}+\frac{1}{1+z}\ge2\)
cmr: \(xyz\le8\)

Answer 1

\(\frac{1}{1+x}\ge1-\frac{1}{1+y}+1-\frac{1}{1+z}=\frac{y}{1+y}+\frac{z}{1+z}\ge2\sqrt{\frac{yz}{\left(1+y\right)\left(1+z\right)}}\)

Tương tự: \(\frac{1}{1+y}\ge2\sqrt{\frac{xz}{\left(1+x\right)\left(1+z\right)}}\) ; \(\frac{1}{1+z}\ge2\sqrt{\frac{xy}{\left(1+x\right)\left(1+y\right)}}\)

Nhân vế với vế:

\(\frac{1}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\ge\frac{8xyz}{\left(1+x\right)\left(1+y\right)\left(1+z\right)}\)

\(\Rightarrow xyz\le\frac{1}{8}< 8\) (đpcm)

Chắc bạn ghi sai đề bài :)