Question

[Trần Phương Thảo]

Tìm giá trị lớn nhất của

-x² -3y² -2xy +10x +16y +18

Answer 1

[Lightning Farron]

\(-x^2 -3y^2 -2xy +10x +16y +18 \)

\(=-x^2-2xy+10x-y^2+10y-25-2y^2+6y+43\)

\(=-\left(x^2+2xy-10x+y^2-10y+25\right)-2\left(y^2-3y-\dfrac{43}{2}\right)\)

\(=-\left[\left(x^2+2xy+y^2\right)-\left(10x+10y\right)+25\right]-2\left(y^2-3y+\dfrac{9}{4}-\dfrac{95}{4}\right)\)

\(=-\left[\left(x+y\right)^2-10\left(x+y\right)+25\right]-2\left(y^2-3y+\dfrac{9}{4}\right)+\dfrac{95}{2}\)

\(=-\left(x+y-5\right)^2-2\left(y-\dfrac{3}{2}\right)^2+\dfrac{95}{2}\le\dfrac{95}{2}\)

Đẳng thức xảy ra khi \(\left\{{}\begin{matrix}-\left(x+y-5\right)^2=0\\-2\left(y-\dfrac{3}{2}\right)^2=0\end{matrix}\right.\)\(\Rightarrow\left\{{}\begin{matrix}x=\dfrac{7}{2}\\y=\dfrac{3}{2}\end{matrix}\right.\)