Question

Tìm x;y;z để gt sau nhỏ nhất

\(\dfrac{1}{\left|x-y\right|+\left|y-z\right|+\left|z-x\right|+2016}\)

Answer 1

Lớn nhất nhé bạn :D

\(A=\dfrac{1}{\left|x-y\right|+\left|y-z\right|+\left|z-x\right|+2016}\)

\(\left\{{}\begin{matrix}\left|x-y\right|\ge0\\\left|y-z\right|\ge0\\\left|z-x\right|\ge0\end{matrix}\right.\) \(\Rightarrow\left|x-y\right|+\left|y-z\right|+\left|z-x\right|\ge0\)

\(\Rightarrow\left|x-y\right|+\left|y-z\right|+\left|z-x\right|+2016\ge2016\)

\(A=\dfrac{1}{\left|x-y\right|+\left|y-z\right|+\left|z-x\right|+2016}\le\dfrac{1}{2016}\)

Dấu "=" xảy ra khi:

\(\left\{{}\begin{matrix}\left|x-y\right|=0\\\left|y-z\right|=0\\\left|z-x\right|=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=y\\y=z\\z=x\end{matrix}\right.\)

\(\Rightarrow x=y=z\)