Question

cho \(0\le x,y,z\le1\) chung minh \(x+y+z-xy-yz-zx\le1\)

Answer 1

Ta có 

x + y + z - xy - yz - xz \(\le1\)

\(\Leftrightarrow\left(1-x\right)+\left(xy-y\right)+\left(yz-xyz\right)+\left(xz-z\right)+xyz\ge0\)

\(\Leftrightarrow\left(1-x\right)\left(1-y-z+yz\right)+xyz\ge0\)

\(\Leftrightarrow\left(1-x\right)\left(\left(1-y\right)+\left(-z+yz\right)\right)+xyz\ge0\)

\(\Leftrightarrow\left(1-x\right)\left(1-y\right)\left(1-z\right)+xyz\ge0\)

Đúng vì theo đề ta có: \(\hept{\begin{cases}1-x\ge0\\1-y\ge0\\1-z\ge0\end{cases}}\)và \(\hept{\begin{cases}x\ge0\\y\ge0\\z\ge0\end{cases}}\)

Vậy ta có ĐPCM