A= 2x^2 + y^2 - 2xy -2x+3

A= x^2-2xy + y^2 + x^2 - 2x+ 1 +2

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

(x-y)^2> hoặc = 0 với mọi giá trị của x

(x-1)^2 > hoặc =0 với mọi giá trị của x

=> (x-y)^2 + (x-1)^2 > hoặc =0 với mọi giá trị của x

=> (x-y)^2 + (x-1)^2 + 2 > hoặc =2

=> A lớn hơn hoặc bằng 2

=> GTNN của A=2 tại x=y=1

\(2N=4x^2+4xy+10y^2-16x-44y+4038\)

\(=4x^2+4x\left(y-4\right)+\left(y-4\right)^2-\left(y-4\right)^2+10y^2-44y+4038\)

\(=\left(2x+y-4\right)^2+9y^2-36y^2+36+3986\)

\(=\left(2x+y-4\right)^2+\left(3y-6\right)^2+3986\ge3986\forall x,y\)

\(\Rightarrow N\ge1993\forall x,y\)

Dấu "=" \(\Leftrightarrow\left\{{}\begin{matrix}\left(2x+y-4\right)^2=0\\\left(3y-6\right)^2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=2\end{matrix}\right.\)

\(M=\dfrac{1}{2}\left(4x^2+y^2+1-4xy+4x-2y\right)+\dfrac{9}{2}y^2+3y-\dfrac{1}{2}\)

\(M=\dfrac{1}{2}\left(2x-y+1\right)^2+\dfrac{9}{2}\left(y+\dfrac{1}{3}\right)^2-1\ge-1\)

\(M_{min}=-1\) khi \(\left\{{}\begin{matrix}x=-\dfrac{2}{3}\\y=-\dfrac{1}{3}\end{matrix}\right.\)

a) \(2x^2-x+1=2\left(x-\dfrac{1}{4}\right)^2+\dfrac{7}{8}\ge\dfrac{7}{8}\)

\(ĐTXR\Leftrightarrow x=\dfrac{1}{4}\)

b) \(5x-x^2+4=-\left(x-\dfrac{5}{2}\right)^2+\dfrac{41}{4}\le\dfrac{41}{4}\)

\(ĐTXR\Leftrightarrow x=\dfrac{5}{2}\)

c) \(x^2+5y^2-2xy+4y+3=\left(x-y\right)^2+\left(2y+1\right)^2+2\ge2\)

\(ĐTXR\Leftrightarrow\)\(x=y=-\dfrac{1}{2}\)

b: ta có: \(-x^2+5x+4\)

\(=-\left(x^2-5x-4\right)\)

\(=-\left(x^2-2\cdot x\cdot\dfrac{5}{2}+\dfrac{25}{4}-\dfrac{41}{4}\right)\)

\(=-\left(x-\dfrac{5}{2}\right)^2+\dfrac{41}{4}\le\dfrac{41}{4}\forall x\)

Dấu '=' xảy ra khi \(x=\dfrac{5}{2}\)

Đặt `A=2x^2+2xy+5y^2-8x-22y`

`<=>2A=4x^2+4xy+10y^2-16x-44y`

`<=>2A=4x^2+4xy+y^2-8(2x+y)+9y^2-28y`

`<=>2A=(2x+y)^2-8(2x+y)+16+9y^2-28y+196/9-196/9`

`<=>2A=(2x+y-4)^2+(3y-14/3)^2-196/9>=-196/9`

`<=>A>=-98/9`

Dấu "=" xảy ra khi `y=14/9,x=(4-y)/2=11/9`

a: Ta có: \(A=x^2+3x+4\)

\(=x^2+2\cdot x\cdot\dfrac{3}{2}+\dfrac{9}{4}+\dfrac{7}{4}\)

\(=\left(x+\dfrac{3}{2}\right)^2+\dfrac{7}{4}\ge\dfrac{7}{4}\forall x\)

Dấu '=' xảy ra khi \(x=-\dfrac{3}{2}\)