\(A=|2x-2|+|2x-2013|\)
\(=|2x-2|+|2013-2x|\ge|2x-2+2013-2x|\)
\(\Rightarrow A\ge2011\)
Dấu "="xảy ra \(\Leftrightarrow\left(2x-2\right)\left(2013-2x\right)\ge0\)
\(\Leftrightarrow\hept{\begin{cases}2x-2\ge0\\2013-2x\ge0\end{cases}}\)hoặc \(\hept{\begin{cases}2x-2< 0\\2013-2x< 0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}x\ge1\\x\le\frac{2013}{2}\end{cases}}\)hoặc \(\hept{\begin{cases}x< 1\\x>\frac{2013}{2}\end{cases}}\)( loại )
\(\Leftrightarrow1\le x\le\frac{2013}{2}\)mà \(x\in Z\)
\(\Rightarrow x\in\left\{1;2;...;1006\right\}\)
Vậy \(A_{min}=2011\)\(\Leftrightarrow x\in\left\{1;2;...;1006\right\}\)
Amin= 2011 với x=1;2;...;1000
