\(P=\left|x-1\right|+\left|2010-x\right|+\sqrt{2019}\)
\(P\ge\left|x-1+2010-x\right|+\sqrt{2019}=2009+\sqrt{2019}\)
\(\Rightarrow P_{min}=2009+\sqrt{2019}\) khi \(\left\{{}\begin{matrix}x-1\ge0\\2010-x\ge0\end{matrix}\right.\) \(\Rightarrow1\le x\le2010\)