Đáp án:
\[{P_{\min }} = 2017 \Leftrightarrow \left\{ \begin{array}{l}
x = - \frac{5}{2}\\
y = - \frac{1}{2}
\end{array} \right.\]
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
P = 3{x^2} + 31{y^2} - 18xy + 6x - 14y + 2021\\
= 3\left[ {\left( {{x^2} - 6xy + 9{y^2}} \right) + 2\left( {x - 3y} \right) + 1} \right] + \left( {4{y^2} + 4y + 1} \right) + 2017\\
= 3\left[ {{{\left( {x - 3y} \right)}^2} + 2\left( {x - 3y} \right) + 1} \right] + {\left( {2y + 1} \right)^2} + 2017\\
= 3{\left( {x - 3y + 1} \right)^2} + {\left( {2y + 1} \right)^2} + 2017 \ge 2017\\
\Rightarrow {P_{\min }} = 2017 \Leftrightarrow \left\{ \begin{array}{l}
x - 3y + 1 = 0\\
2y + 1 = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
x = - \frac{5}{2}\\
y = - \frac{1}{2}
\end{array} \right.
\end{array}\)
