Ta có :
\(N=\left|x-2014\right|+\left|2015-x\right|\ge\left|x-2014+2015-x\right|=1\)
Dấu "=" xảy ra khi :
\(\Leftrightarrow\left(x-2014\right)\left(2015-x\right)\ge0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-2014\ge0\\2015-x\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}x-2014\le0\\2015-x\le0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x\ge2014\\2015\ge x\end{matrix}\right.\\\left\{{}\begin{matrix}x\le2014\\2015\le x\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}2014\le x\le2015\\x\in\varnothing\end{matrix}\right.\)
Vậy \(N_{Min}=1\Leftrightarrow2014\le x\le2015\)
N= | x-2014 | +|2015 -x| ≥| x-2014 + 2015 -x | = | 1| = 1
dấu "=" khi x-2014 = 2015 - x
<=> x = 2014,5
vậy gtnn N = 1 khi x = 2014,5
