Toán Lớp 8: Tìm max: H= -x^2 + xy – y^2 – 2x + 4y + 11
A = 5 – 2x^2 – 4y^2 + 4xy – 8x – 12y
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Toán Lớp 8: Tìm max: H= -x^2 + xy – y^2 – 2x + 4y + 11
A = 5 – 2x^2 – 4y^2 + 4xy – 8x – 12y
Comments ( 1 )
1)\quad H = -x^2 + xy – y^2 – 2x + 4y + 11\\
\to 4H = -(4x^2 – 4xy + y^2 + 8x – 4y + 4) – (3y^2 – 12y +12) + 60\\
\to 4H = -(2x – y + 2)^2 – 3(y – 2)^2 + 60\\
\to 4H \geqslant 60\\
\to H \geqslant 15\\
\text{Dấu = xảy ra}\ \Leftrightarrow \begin{cases}2x -y + 2 =0\\y – 2 =0\end{cases}\Leftrightarrow \begin{cases}x = 0\\y = 2\end{cases}\\
\text{Vậy}\ \max H = 15 \Leftrightarrow (x;y) = (0;2)\\
2)\quad A = 5 – 2x^2 -4y^2 + 4xy – 8x – 12y\\
\to A = -(2x^2 – 4xy + y^2 + 8x – 8y + 8) – (2y^2 + 20y + 50) +63\\
\to A = -2(x – y + 2)^2 – 2(y + 5)^2 + 63\\
\to A \geqslant 63\\
\text{Dấu = xảy ra}\ \Leftrightarrow \begin{cases}x – y + 2 =0\\y + 5 =0\end{cases}\Leftrightarrow \begin{cases}x = -7\\y = -5\end{cases}\\
\text{Vậy}\ \max A = 63 \Leftrightarrow (x;y) = (-7;-5)
\end{array}\)