Question

[deco]

tìm GTLN:

-2x²-y²+4x-2xy+20

Answer 1

[Toru]

\(\text{Đặt }A=-2x^2-y^2+4x-2xy+20\\=-(x^2+2xy+y^2)-(x^2-4x+4)+24\\=-(x+y)^2-(x-2)^2+24\)

Ta thấy: \(\left\{{}\begin{matrix}\left(x+y\right)^2\ge0;\forall x,y\\\left(x-2\right)^2\ge0;\forall x\end{matrix}\right.\Rightarrow\left(x+y\right)^2+\left(x-2\right)^2\ge0;\forall x,y\)

\(\Rightarrow-\left(x+y\right)^2-\left(x-2\right)^2\le0;\forall x,y\)

\(\Rightarrow-\left(x+y\right)^2-\left(x-2\right)^2+24\le24;\forall x,y\)

\(\Rightarrow A\le24;\forall x,y\)

Dấu "=" xảy ra khi: \(\left\{{}\begin{matrix}x+y=0\\x-2=0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x=2\\y=-2\end{matrix}\right.\)

Vậy: ...