Question

Tìm x ∈ Z để biểu thức

|x-5| + |x-7| đạt GTNN

Answer 1

\(\left|x-5\right|+\left|x-7\right|\\ =\left|5-x\right|+\left|x-7\right|\\ \ge\left|5-x+x-7\right|\\ =\left|-2\right|\\ =2\)

Dấu "=" xảy ra \(\Leftrightarrow\left(5-x\right)\left(x-7\right)\ge0\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}5-x\ge0\\x-7\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}5-x\le0\\x-7\le0\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\le5\\x\ge7\left(vô.lí\right)\end{matrix}\right.\\\left\{{}\begin{matrix}x\ge5\\x\le7\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow5\le x\le7\)

Vậy \(5\le x\le7\) thì \(\left|x-5\right|+\left|x-7\right|\) đạt GTNN