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

Do \(\left\{{}\begin{matrix}\left(x-y\right)^2\ge0\\\left(y+1\right)^2\ge0\end{matrix}\right.\) ;\(\forall x;y\)

\(\Rightarrow\left(x-y\right)^2+\left(y+1\right)^2+4>0\) ; \(\forall x;y\)

ko biết

Bài 2:

Ta có: M = a²+ab+b² -3a-3b-3a-3b +2001

=> 2M = ( a² + 2ab + b²) -4.(a+b) +4 + (a² -2a+1)+(b² -2b+1) + 3996

2M= ( a+b-2)² + (a-1)² +(b-1)² + 3996

=> Min_M = 1998 tại a=b=1

Câu 3: 

Ta có: P= x² +xy+y² -3.(x+y) + 3

=> 2P = ( x² + 2xy +y²) -4.(x+y) + 4 + (x² -2x+1) +(y² -2y+1)

2P = ( x+y-2)² +(x-1)²+(y-1)²

=> Min_P = 0 tại x=y=1

ta co x2+3y2=4xy suy ra x2+3y2-4xy=0 suy ra x2-xy-3xy+3y2=0 suy ra x(x-y)-3y(x-y)=0 suy ra (x-3y)(x-y)=0

​với x-y=0 suy ra x=y mà theo đề bài x>y>0 suy ra x-3y=0 suy ra x=3y thay vào P là xong 

Ban coi co dung khong nha

Bạn thay như thế nào thế ? Bạn làm luôn được không ? 

\(5x^2+2y^2+6xy-8x-4y+4=0\)

\(\Leftrightarrow4x^2+x^2+y^2+y^2+2xy+4xy-8x-4y+4=0\)

\(\Leftrightarrow\left(4x^2+y^2+4+4xy-8x-4y\right)+\left(x^2+2xy+y^2\right)=0\)

\(\Leftrightarrow\left[\left(2x\right)^2+4xy+y^2-4\left(2x+y\right)+2^2\right]+\left(x+y\right)^2=0\)

\(\Leftrightarrow\left[\left(2x+y\right)^2-2\cdot\left(2x+y\right)\cdot2+2^2\right]+\left(x+y\right)^2=0\)

\(\Leftrightarrow\left(2x+y-2\right)^2+\left(x+y\right)^2=0\)

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

\(\Rightarrow\left(2x+y-2\right)^2+\left(x+y\right)^2\ge0\forall x,y\)

Mặt khác: \(\left(2x+y-2\right)^2+\left(x+y\right)^2=0\) 

Dấu "=" xảy ra khi:

\(\left\{{}\begin{matrix}2x+y-2=0\\x+y=0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2\cdot\left(-y\right)+y-2=0\\x=-y\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-2y+y-2=0\\x=-y\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-y=2\\x=-y\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}y=-2\\x=2\end{matrix}\right.\) 

Thay x,y vào P ta có:

\(P=2^{2023}+\left(-2\right)^{2023}=2^{2023}-2^{2023}=0\)

Vậy: ... 

`B = x^2- 2xy + y^2 + 2x - 10y + 17

`2B = 2x^2 - 4xy + 2y^2 + 4x - 20y + 34`

`= (x-y)^2 + (x+2)^2 + (y-5)^2 + 5 >= 5`.

Mik cảm ơn