Question

[Bui Thu Phuong]

Tìm GTLN của F = - x^2 - 2y^2 + 2xy - y +1  Mn giúp em vs ạk💓

Answer 1

[hya_seije_jaumeniz]

\(F=-x^2-2y^2+2xy-y+1\)

\(-F=x^2+2y^2-2xy+y-1\)

\(-F=\left(x^2-2xy+y^2\right)+\left(y^2+y+\frac{1}{4}\right)-\frac{5}{4}\)

\(-F=\left(x-y\right)^2+\left(y+\frac{1}{2}\right)^2-\frac{5}{4}\)

Mà  \(\left(x-y\right)^2\ge0\forall x;y\)

      \(\left(y+\frac{1}{2}\right)^2\ge0\forall y\)

\(\Rightarrow-F\ge-\frac{5}{4}\)

\(\Leftrightarrow F\le\frac{5}{4}\)

Dấu "=" xảy ra khi : 

\(\hept{\begin{cases}x-y=0\\y+\frac{1}{2}=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=-\frac{1}{2}\\y=-\frac{1}{2}\end{cases}}\)

Vậy \(F_{Max}=\frac{5}{4}\Leftrightarrow x=y=-\frac{1}{2}\)

Answer 2

[Bui Thu Phuong]

Hihii tks ban nhieu nha <3