Tìm giá trị nhỏ nhất a)x^2+4y^2-2xy-6y-10.(x-y)+32 b)y^2-2xy+3x^2+2y-14x+2020 c)2x^2+2xy+5y^2-3x-22y

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1. Đáp án:

   $\begin{array}{l}
   a)A = {x^2} + 4{y^2} – 2xy – 6y – 10\left( {x – y} \right) + 32\\
    = {x^2} – 2xy + {y^2} – 10\left( {x – y} \right) + 3{y^2} – 6y + 32\\
    = {\left( {x – y} \right)^2} – 10\left( {x – y} \right) + 3\left( {{y^2} – 2y + 1} \right) + 29\\
    = {\left( {x – y} \right)^2} – 10\left( {x – y} \right) + 25 + 3{\left( {y – 1} \right)^2} + 4\\
    = {\left( {x – y – 5} \right)^2} + 3{\left( {y – 1} \right)^2} + 4\\
   Do:\left\{ \begin{array}{l}
   {\left( {x – y – 5} \right)^2} \ge 0\\
   3{\left( {y – 1} \right)^2} \ge 0
   \end{array} \right.\\
    \Rightarrow A \ge 4\\
    \Rightarrow GTNN:A = 4\\
   Khi:\left\{ \begin{array}{l}
   x – y – 5 = 0\\
   y – 1 = 0
   \end{array} \right.\\
    \Rightarrow \left\{ \begin{array}{l}
   x = 6\\
   y = 1
   \end{array} \right.\\
   b){y^2} – 2xy + 3{x^2} + 2y – 14x + 2020\\
    = {y^2} + {x^2} + 1 – 2xy + 2y – 2x + 2{x^2} – 12x + 2019\\
    = {\left( {y – x + 1} \right)^2} + 2\left( {{x^2} – 6x + 9} \right) + 2001\\
    = {\left( {y – x + 1} \right)^2} + 2{\left( {x – 3} \right)^2} + 2001\\
    \ge 2001\\
    \Rightarrow GTNN = 2001\\
   Khi:\left\{ \begin{array}{l}
   y – x + 1 = 0\\
   x – 3 = 0
   \end{array} \right.\\
    \Rightarrow \left\{ \begin{array}{l}
   y = 2\\
   x = 3
   \end{array} \right.\\
   c)C = 2{x^2} + 2xy + 5{y^2} – 3x – 22y\\
    = {x^2} + 2xy + {y^2} + {x^2} – 3x + \dfrac{9}{4}\\
    + 4{y^2} – 22y – \dfrac{9}{4}\\
    = {\left( {x + y} \right)^2} + {\left( {x – \dfrac{3}{2}} \right)^2}\\
    + 4.\left( {{y^2} – \dfrac{{11}}{2}y + \dfrac{{121}}{{16}}} \right) – \dfrac{{65}}{2}\\
    = {\left( {x + y} \right)^2} + {\left( {x – \dfrac{3}{2}} \right)^2} + 4{\left( {y – \dfrac{{11}}{4}} \right)^2} – \dfrac{{65}}{2}\\
    \Rightarrow GTNN:C = \dfrac{{ – 65}}{2}
   \end{array}$

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