Toán Lớp 8: Tìm min: E= x^2 – xy + 3y^2 – 2x – 10y -3
E= -x^2 + 2xy – 4y^2 + 2x + 10y – 3
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Toán Lớp 8: Tìm min: E= x^2 – xy + 3y^2 – 2x – 10y -3
E= -x^2 + 2xy – 4y^2 + 2x + 10y – 3
Comments ( 1 )
1)\quad E = x^2 – xy +3y^2 -2x – 10y – 3\\
\to 4E = 4x^2 – 4xy + 12y^2 – 8x – 40y – 12\\
\to 4E = (4x^2 – 4xy + y^2 -8x+ 4y+ 4) + (11y^2 – 44y +44)-60\\
\to 4E = (2x – y – 2)^2 + 11(y – 2)^2 – 60\\
\to 4E \geqslant -60\\
\to E \geqslant -15\\
\text{Dấu = xảy ra}\ \Leftrightarrow \begin{cases}2x – y – 2 = 0\\y – 2=0\end{cases}\Leftrightarrow x = y = 2\\
\text{Vậy}\ \min E = -15 \Leftrightarrow (x;y)= (2;2)\\
2)\quad E = -x^2 + 2xy – 4y^2 + 2x + 10y – 3\\
\to E = -(x^2 – 2xy + y^2 -2x + 2y +1) – (3y^2 – 12y + 12) + 10\\
\to E = -(x – y – 1)^2 – 3(y-2)^2 + 10\\
\to E \leqslant 10\\
\text{Dấu = xảy ra}\ \Leftrightarrow \begin{cases}x – y – 1 =0\\y – 2 =0\end{cases}\Leftrightarrow \begin{cases}x = 3\\y = 2\end{cases}\\
\text{Vậy}\ \max E = 10\Leftrightarrow (x;y) = (3;2)
\end{array}\)