\(A=\left|x-2009\right|+\left|x-3\right|\)
\(A=\left|x-2009\right|+\left|3-x\right|\)
\(A\ge\left|x-2009+3-x\right|\)
\(A\ge2006\)
Dấu "=" xảy ra khi:
\(\left[{}\begin{matrix}\left\{{}\begin{matrix}x-2009\ge0\Rightarrow x\ge2009\\3-x\ge0\Rightarrow x\le3\end{matrix}\right.\\\left\{{}\begin{matrix}x-2009\le0\Rightarrow x\le2009\\3-x\le0\Rightarrow x\ge3\end{matrix}\right.\end{matrix}\right.\)
Suy ra \(3\le x\le2009\)
Ta có \(|x-2019|\ge2019-x\)Với mọi x
\(\left|x-3\right|\ge x-3\)Với mọi x
\(\Rightarrow\left|x-2019\right|+\left|x-3\right|\ge2019-x+x-3=2016\) Với mọi x
\(\)\(\)Amin \(=\)2016 khi \(x-2019\le0\) và \(x-3\ge0\)
\(\Rightarrow\)Amin \(=\)2016 khi \(x\le2019\) và \(x\ge3\)
Vậy Amin =2016 khi \(1\le x\le2019\)
