a) \(5x^2-12xy+9y^2-4x+4=\left(4x^2-12xy+9y^2\right)+x^2-4x+4=\left(2x-3y\right)^2+\left(x-2\right)^2\ge0\)
b) \(-x^2-2y^2+12x-4y+7=-\left(x^2-12x+36\right)-2\left(y^2+2y+1\right)+45=-\left(x-6\right)^2-2\left(y+1\right)^2+45\le45\)

c)\(4y^2+10x^2+12xy+6x+7=\left(4y^2+12xy+9x^2\right)+x^2+6x+9-2=\left(2y+3x\right)^2+\left(x+3\right)^2-2\ge-2\)

d) \(3-10x^2-4xy-4y^2=3-\left(4y^2+4xy+x^2\right)-9x^2=-\left(2y+x\right)^2-9x^2+3\le3\)

e)\(x^2-5x+y^2-xy-4y+16=\left(\frac{1}{2}x^2-xy+\frac{1}{2}y^2\right)+\frac{1}{2}\left(x^2-10x+25\right)+\frac{1}{2}\left(y^2-8y+16\right)-\frac{9}{2}=\frac{1}{2}\left(x-y\right)^2+\frac{1}{2}\left(x-5\right)^2+\frac{1}{2}\left(y-4\right)^2-\frac{9}{2}\ge-\frac{9}{2}\)Phần e) mới nghĩ đk v, tui biết đáp án sao do k xảy ra dấu bằng

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

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

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

MinM=-4

Dấu "=" xảy ra khi \(\hept{\begin{cases}2x-y=0\\y-2=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=1\\y=2\end{cases}}}\)

\(A=\left(2x-1\right)^2+9\ge9\\ A_{min}=9\Leftrightarrow x=\dfrac{1}{2}\\ B=2\left(x^2-2\cdot\dfrac{3}{4}x+\dfrac{9}{16}\right)+\dfrac{1}{8}=2\left(x-\dfrac{3}{4}\right)^2+\dfrac{1}{8}\ge\dfrac{1}{8}\\ B_{min}=\dfrac{1}{8}\Leftrightarrow x=\dfrac{3}{4}\\ C=\left(4x^2+4xy+y^2\right)+2\left(2x+y\right)+1+\left(y^2+4y+4\right)-4\\ C=\left[\left(2x+y\right)^2+2\left(2x+y\right)+1\right]+\left(y+2\right)^2-4\\ C=\left(2x+y+1\right)^2+\left(y+2\right)^2-4\ge-4\\ C_{min}=-4\Leftrightarrow\left\{{}\begin{matrix}2x=-1-y\\y=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{3}{2}\\y=-2\end{matrix}\right.\)

\(D=\left(3x-1-2x\right)^2=\left(x-1\right)^2\ge0\\ D_{min}=0\Leftrightarrow x=1\\ G=\left(9x^2+6xy+y^2\right)+\left(y^2+4y+4\right)+1\\ G=\left(3x+y\right)^2+\left(y+2\right)^2+1\ge1\\ G_{min}=1\Leftrightarrow\left\{{}\begin{matrix}3x=-y\\y=-2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=\dfrac{2}{3}\\y=-2\end{matrix}\right.\)

\(H=\left(x^2-2xy+y^2\right)+\left(x^2+2x+1\right)+\left(2y^2+4y+2\right)+2\\ H=\left(x-y\right)^2+\left(x+1\right)^2+2\left(y+1\right)^2+2\ge2\\ H_{min}=2\Leftrightarrow\left\{{}\begin{matrix}x=y\\x=-1\\y=-1\end{matrix}\right.\Leftrightarrow x=y=-1\)

Ta luôn có \(\left(x-y\right)^2+\left(y-z\right)^2+\left(z-x\right)^2\ge0\)

\(\Leftrightarrow2x^2+2y^2+2z^2-2xy-2yz-2xz\ge0\\ \Leftrightarrow x^2+y^2+z^2\ge xy+yz+xz\\ \Leftrightarrow x^2+y^2+z^2+2xy+2yz+2xz\ge3xy+3yz+3xz\\ \Leftrightarrow\left(x+y+z\right)^2\ge3\left(xy+yz+xz\right)\\ \Leftrightarrow\dfrac{3^2}{3}\ge xy+yz+xz\\ \Leftrightarrow K\le3\\ K_{max}=3\Leftrightarrow x=y=z=1\)

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

\(C_{min}=-\dfrac{1}{2}\) khi \(x=\dfrac{1}{2}\)

\(D=\left(16x^2+2x+\dfrac{1}{16}\right)-\dfrac{1}{16}=\left(4x+\dfrac{1}{4}\right)^2-\dfrac{1}{16}\ge-\dfrac{1}{16}\)

\(D_{min}=-\dfrac{1}{16}\) khi \(x=-\dfrac{1}{16}\)

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

\(E=\left(x-2y\right)^2+\left(2x-1\right)^2+2\ge2\)

\(E_{min}=2\) khi \(\left(x;y\right)=\left(\dfrac{1}{2};\dfrac{1}{4}\right)\)

a) Ta có A = x² - 2x - 1 = (x² - 2x + 1) - 2 = (x - 1)² - 2 \(\ge\) -2 

Dấu "=" xảy ra <=> x - 1 = 0 => x = 1

Vậy Min A = -2 <=> x = 1 

b) Ta có B = 4x² + 4x + 8 = (4x² + 4x + 1) + 7 = (2x + 1)² + 7 \(\ge\)7

Dấu |"=" xảy ra <=> 2x + 1 = 0 => x = -1/2

Vậy Min B = 7 <=> x = -1/2

c) Ta có C = 3x - x² + 2

                 = -(x² - 3x - 2)

                = -(x² - 3x + 9/4 - 9/4 - 2)

                = -[(x - 3/2)² - 17/4)

                 = -(x - 3/2)² + 17/4 \(\le\frac{17}{4}\)

Dấu "=" xảy ra <=> x - 3/2 = 0 => x = 3/2

Vậy Max C = 17/4 <=> x = 3/2

d) Ta có D = -x² - 5x = -(x² + 5x) = -(x² + 5x + 25/4 - 25/4) = -(x + 5/2)² + 25/4 \(\ge\frac{25}{4}\)

Dấu "=" xảy ra <=> x + 5/2 = 0 => x = -5/2

Vậy Max D = 25/4 <=> x = -5/2

e) Ta có E = x² - 4xy + 5y² + 10x - 22y + 28

                  = (x² - 4xy + 4y²) + 10x - 20y + y² - 2y + 28

                 = (x - 2y)² + 10(x - 2y) + 25 + (y² - 2y + 1) + 2

                 = (x - 2y + 5) + (y - 1)² + 2 \(\ge\)2

Dấu "=" xảy ra <=> \(\hept{\begin{cases}x-2y+5=0\\y-1=0\end{cases}}\Rightarrow\hept{\begin{cases}x=-3\\y=1\end{cases}}\)

Vậy Min E = 2 <=> x = -3 ; y = 1