Toán Lớp 8: Tìm min: K = x^2 + y^2 – xy + 3x + 3y + 20
A = a^2 + ab + b^2 – 3a – 3b + 3
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Toán Lớp 8: Tìm min: K = x^2 + y^2 – xy + 3x + 3y + 20
A = a^2 + ab + b^2 – 3a – 3b + 3
Comments ( 1 )
1)\quad K = x^2 + y^2 – xy + 3x + 3y + 20\\
\to 4K = (4x^2 – 4xy + y^2 + 12x – 6y + 9) + (3y^2 + 18y + 27) + 44\\
\to 4K = (2x – y +3)^2 + 3(y + 3)^2 + 44\\
\to 4K \geqslant 44\\
\to K \geqslant 11\\
\text{Dấu = xảy ra}\ \Leftrightarrow \begin{cases}2x – y + 3 =0\\y + 3=0\end{cases}\Leftrightarrow x = y = -3\\
\text{Vậy}\ \min K = 11 \Leftrightarrow (x;y) = (-3;-3)\\
2)\quad A = a^2 + ab + b^2 – 3a – 3b + 3\\
\to 4A = (4a^2 + 4ab + b^2 – 12a – 6b + 9) + (3b^2 – 6b +3)\\
\to 4A = (2a + b – 3)^2 + 3(b – 1)^2\\
\to 4A \geqslant 0\\
\to A \geqslant 0\\
\text{Dấu = xảy ra}\ \Leftrightarrow \begin{cases}2a + b – 3 =0\\b – 1 =0\end{cases}\Leftrightarrow a = b = 1\\
\text{Vậy}\ \min A = 0\Leftrightarrow (a;b) = (1;1)
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