\(\left|x-2\right|+\left|x-4\right|\)
\(=\left|2-x\right|+\left|x-4\right|\ge\left|2-x+x-4\right|\)\(=2\)
\(\Rightarrow\left|x-2\right|+\left|x-4\right|\ge2\)
Dấu" = "xảy ra khi:\(\left(2-x\right).\left(x-4\right)\)\(\ge0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}2-x\le0\\x-4\le0\end{matrix}\right.\\\left\{{}\begin{matrix}2-x\ge0\\x-4\ge0\end{matrix}\right.\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\le2\\x\ge4\end{matrix}\right.\\\left\{{}\begin{matrix}x\ge2\\x\le4\end{matrix}\right.\end{matrix}\right.\)
\(\left\{{}\begin{matrix}x\le2\\x\ge4\end{matrix}\right.\)(ko có số nào thỏa mãn)
\(\left\{{}\begin{matrix}x\ge2\\x\le4\end{matrix}\right.\)\(\Rightarrow2\le x\le4\)
Vậy \(2\le x\le4\)
\(\left|x-2\right|+\left|x-4\right|\)
\(=\left|2-x\right|+\left|x-4\right|\ge\left|2-x+x-4\right|=\left|-2\right|=2\)
\(\Rightarrow\left|x-2\right|+\left|x-4\right|\ge2\)
Dấu \("="\) xảy ra khi: \(\left(2-x\right)\left(x-4\right)\ge0\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left\{{}\begin{matrix}2-x\ge0\\x-4\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}2-x\le0\\x-4\le0\end{matrix}\right.\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}\left\{{}\begin{matrix}x\le2\left(loại\right)\\x\ge4\left(loại\right)\end{matrix}\right.\\\left\{{}\begin{matrix}x\ge2\left(TM\right)\\x\le4\left(TM\right)\end{matrix}\right.\end{matrix}\right.\)
\(\Leftrightarrow2\le x\le4\)
Vậy GTNN của \(\left|x-2\right|+\left|x-4\right|\) là 1 khi \(2\le x\le4\)
