Toán Lớp 8: tìm gtln:
F = 4 – 5x^2 – 2y^2 + 6xy – 4x + 2y
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Toán Lớp 8: tìm gtln:
F = 4 – 5x^2 – 2y^2 + 6xy – 4x + 2y
Comments ( 1 )
\quad F = 4 – 5x^2 – 2y^2 + 6xy – 4x + 2y\\
\to 5F = -25x^2 – 10y^2 +30xy -20x + 10y + 20\\
\to 5F = -(25x^2 – 30xy + 9y^2 + 20x -12y + 4) – (y^2+2y+ 1) +25\\
\to 5F = -(5x – 3y + 2)^2 – (y+1)^2 + 25\\
\text{Ta có:}\\
\quad \begin{cases}(5x – 3y + 2)^2 \geqslant 0\quad \forall x;y\\(y+1)^2 \geqslant 0\quad \forall y\end{cases}\\
\Leftrightarrow \begin{cases}-(5x – 3y + 2) \leqslant 0\\-(y+1)^2 \leqslant 0\end{cases}\\
\Leftrightarrow -(5x – 3y + 2)^2 – (y+1)^2 + 25 \leqslant 25\\
\Leftrightarrow 5F \leqslant 25\\
\Leftrightarrow F \leqslant 5\\
\text{Dấu = xảy ra}\ \begin{cases}5x – 3y + 2 =0\\y + 1 =0\end{cases}\Leftrightarrow x = y= -1\\
\text{Vậy}\ \max F = 5 \Leftrightarrow (x;y)= (-1;-1)
\end{array}\)