Đáp án: max $P = \frac{{{{2019}^2}}}{4}$
Giải thích các bước giải:
$\begin{array}{l}
x + y + z = 2019\\
\Rightarrow x + y = 2019 - z\\
\Rightarrow P = \left( {x + y} \right)z\\
= \left( {2019 - z} \right).z\\
Áp\,dụng\,bdt:a.b \le \frac{{{{\left( {a + b} \right)}^2}}}{4}\\
\Rightarrow \left( {2019 - z} \right).z \le \frac{{{{\left( {2019 - z + z} \right)}^2}}}{4} = \frac{{{{2019}^2}}}{4}\\
\Rightarrow P \le \frac{{{{2019}^2}}}{4}\\
Dấu = xảy\,ra \Leftrightarrow 2019 - z = z \Rightarrow \left\{ \begin{array}{l}
z = \frac{{2019}}{2}\\
x + y = \frac{{2019}}{2}
\end{array} \right.
\end{array}$
Vậy max $P = \frac{{{{2019}^2}}}{4}$
