Lời giải:
$A=x^2+2x+2xy+2y^2+4y+2021$

$=(x^2+2xy+y^2)+2x+y^2+4y+2021$

$=(x+y)^2+2(x+y)+1+(y^2+2y+1)+2019$

$=(x+y+1)^2+(y+1)^2+2019\geq 2019$

Vậy $A_{\min}=2019$ khi $x+y+1=y+1=0$

$\Leftrightarrow (x,y)=(0,-1)$

Đặt `A=2x^2+2y^2+2xy-4x+4y+2021`

`<=>2A=4x^2+4y^2+4xy-8x+8y+4042`

`<=>2A=4x^2+4xy+y^2-8x-4y+3y^2+12y+4042`

`<=>2A=(2x+y)^2-4(2x+y)+4+3y^2+12y+12+4026`

`<=>2A=(2x+y-2)^2+3(y+2)^2+4026>=4026`

`=>A>=2013`

Dấu "=" xảy ra khi `y=-2,x=(2-y)/2=2`

D ez nhất :v

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

\(=\left(x-1\right)^2+\left(y+2\right)^2+5\ge5\)

Đẳng thức xảy ra khi x = 1 và y = -2

\(A=\left[\left(x^2-2xy+y^2\right)+4\left(x-y\right)+4\right]+\left(y^2-2y+1\right)+2020\)

\(=\left[\left(x-y\right)^2+2\left(x-y\right).2+2^2\right]+\left(y-1\right)^2+2020\)

\(=\left(x-y+2\right)^2+\left(y-1\right)^2+2020\ge2020\)

Dấu "=" xảy ra khi y = 1 và x - y + 2 = 0 tức là x = y - 2 = -1

\(B=\left(x^2-2xy+y^2\right)-2\left(x-y\right)+1+x^2-2x+1+2019\)

\(=\left(x-y\right)^2-2\left(x-y\right).1+1+\left(x-1\right)^2+2019\)

\(=\left(x-y-1\right)^2+\left(x-1\right)^2+2019\ge2019\)

Dấu "=" xảy ra khi x = 1 và x - y - 1 = 0 hay y = 0

câu a hình như sai, đúng ra phải là 2x^2 chứ nhỉ, theo đề tính ra thì thừa 2x

câu b nhỏ nhất = 2014, cần cách làm ko z

Nếu được bạn cho mình cách giải đi ạ!

a, 2x² + 2y² + 2xy - 6y - 2x + 2021

= x² + 2xy + y² + y² - 2 * 3y + 9 + x² - 2x + 1 + 2011

= (x + y)² + (y + 3)² + (x - 1)² + 2011

=> GTNN = 2011

Lời giải:

$2x^2-2xy-4y^2=2(x^2-xy-2y^2)$

$=2[(x^2-2xy)+(xy-2y^2)]$

$=2[x(x-2y)+y(x-2y)]$

$=2(x+y)(x-2y)$

-----------------

$x^2-2x-4y^2-4y=(x^2-2x+1)-(4y^2+4y+1)$

$=(x-1)^2-(2y+1)^2=(x-1-2y-1)(x-1+2y+1)$

$=(x-2y-2)(x+2y)$
-------------------

$x^2-4y^2-x-2y=(x^2-4y^2)-(x+2y)=(x-2y)(x+2y)-(x+2y)$

$=(x+2y)(x-2y-1)$

a) \(3x^2-3xy-5x+5y\)

\(=\left(3x^2-3xy\right)-\left(5x-5y\right)\)

\(=3x\left(x-y\right)-5\left(x-y\right)\)

\(=\left(x-y\right)\left(3x-5\right)\)

b) \(2x^3y-2xy^3-4xy^2-2xy\)

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

\(=2xy\left[x^2-\left(y^2+2y+1\right)\right]\)

\(=2xy\left[x^2-\left(y+1\right)^2\right]\)

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

c) \(x^2+1+2x-y^2\)

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

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

\(=\left(x+1+y\right)\left(x+1-y\right)\)

d) \(x^2+4x-2xy-4y+y^2\)

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

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

\(=\left(x-y\right)\left(x-y+4\right)\)

e) \(x^3-2x^2+x\)

\(=x\left(x^2-2x+1\right)\)

\(=x\left(x-1\right)^2\)

f) \(2x^2+4x+2-2y^2\)

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

\(=2\left[\left(x^2+2x+1\right)+y^2\right]\)

\(=2\left[\left(x+1\right)^2-y^2\right]\)

\(=2\left(x-y+1\right)\left(x+y+1\right)\)

a: =3x(x-y)-5(x-y)

=(x-y)(3x-5)

b: \(=2xy\left(x^2-y^2-2y-1\right)\)

\(=2xy\left[x^2-\left(y^2+2y+1\right)\right]\)

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

d:

Sửa đề: x^2+4x-2xy-4y+y^2

=x^2-2xy+y^2+4x-4y

=(x-y)^2+4(x-y)

=(x-y)(x-y+4)

e: =x(x^2-2x+1)

=x(x-1)^2

f: =2(x^2+2x+1-y^2)

=2[(x+1)^2-y^2]

=2(x+1+y)(x+1-y)

a)\(M=x^2-2xy+2y^2-4y+2016\)

\(=\left(x^2-2xy+y^2\right)+\left(y^2-4y+4\right)+2012\)

\(=\left(x-y\right)^2+\left(y-2\right)^2+2012\ge2012\)

Dấu = khi \(\begin{cases}\left(x-y\right)^2=0\\\left(y-2\right)^2=0\end{cases}\)\(\Leftrightarrow\begin{cases}x-y=0\\y-2=0\end{cases}\)

\(\Leftrightarrow\begin{cases}x=y\\y=2\end{cases}\)\(\Leftrightarrow x=y=2\)

Vậy Min_M=2012 khi x=y=2

b)\(N=x^2-2xy+2x+2y^2-4y+2016\)

\(=\left(x^2-2xy+2x+y^2-2y+1\right)+\left(y^2-2y+1\right)+2014\)

\(=\left(x-y+1\right)^2+\left(y-1\right)^2+2014\ge2014\)

Dấu = khi \(\begin{cases}\left(x-y+1\right)^2=0\\\left(y-1\right)^2=0\end{cases}\)\(\Leftrightarrow\begin{cases}x-y+1=0\\y-1=0\end{cases}\)

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

Vậy Min_N=2014 khi x=0;y=1