Đáp án đúng: D
Giải chi tiết:\(A = \frac{{2009}}{{{{\left( {x - 4} \right)}^2} + 7}}\)
Vì \({\left( {x - 4} \right)^2} \ge 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\forall x \in \mathbb{Z}} \right)\)
\( \Rightarrow {\left( {x - 4} \right)^2} + 7 \ge 7\,\,\,\,\,\,\,\,\,\,\,\,\left( {\forall x \in \mathbb{Z}} \right)\)
\( \Rightarrow \frac{{2019}}{{{{\left( {x - 4} \right)}^2} + 7}} \le \frac{{2019}}{7}\,\,\,\left( {\forall x \in \mathbb{Z}} \right)\)
\( \Rightarrow A \le \frac{{2019}}{7}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {\forall x \in \mathbb{Z}} \right)\)
Dấu “\( = \)” xảy ra khi \({\left( {x - 4} \right)^2} = 0\, \Rightarrow x - 4 = 0 \Rightarrow x = 4\).
Vậy \({\rm{max}}\,A = \frac{{2019}}{7}\) tại \(x = 4.\)
Chọn D.