\(x^2+9y^2+4z^2-2x+12y-4z+20=0\)

\(\Leftrightarrow\left(x^2-2x+1\right)+\left(9y^2+12y+4\right)+\left(4z^2-4z+1\right)+14=0\)

\(\Leftrightarrow\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14=0\)(1)

Ta thấy\(\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14\ge14>0\forall x;y;z\)

Nên dấu (1) không thể xảy ra , Hay \(x;y;z\) ko tồn tại (đpcm)

\(a,\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(x^2+x+\dfrac{1}{4}\right)+\dfrac{7}{4}=0\\ \Leftrightarrow\left(x-y\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{7}{4}=0\\ \Leftrightarrow x,y\in\varnothing\left[\left(x-y\right)^2+\left(x+\dfrac{1}{2}\right)^2+\dfrac{7}{4}\ge\dfrac{7}{4}>0\right]\\ b,\Leftrightarrow\left(x^2-2x+1\right)+\left(9y^2+12y+4\right)+\left(4z^2-4z+1\right)+14=0\\ \Leftrightarrow\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14=0\\ \Leftrightarrow x,y,z\in\varnothing\left[\left(x-1\right)^2+\left(3y+2\right)^2+\left(2z-1\right)^2+14\ge14>0\right]\)

\(c,\Leftrightarrow-\left(x^2-10xy+25y^2\right)-\left(y^2-20y+100\right)-50=0\\ \Leftrightarrow-\left(x-5y\right)^2-\left(y-10\right)^2-50=0\\ \Leftrightarrow x,y\in\varnothing\left[-\left(x-5y\right)^2-\left(y-10\right)^2-50\le-50< 0\right]\)

a) = (x² - 2xy +y²) + (x² +x +2)

=(x-y)² + (x+1/2)² +7/4 >0 với mọi x,y

=> không tồn tại các số x,y thỏa mãn hằng đẳng thức đã cho.

b) = (x²-2x+1)+(9y²+12y+4)+(4z²-4z+1) + 14=(x-1)²+(3y+2)²+(2z+1)²+14>0 với mọi x,y ,z

=> không tồn tại giá trị x,y,z thỏa mãn đẳng thức đã cho

Bài 1:

$A=2x^2+y^2-2xy+x+2=(x^2+y^2-2xy)+(x^2+x+\frac{1}{4})+\frac{7}{4}$

$=(x-y)^2+(x+\frac{1}{2})^2+\frac{7}{4}$

Vì $(x-y)^2\geq 0; (x+\frac{1}{2})^2\geq 0$ với mọi $x,y$

$\Rightarrow A\geq 0+0+\frac{7}{4}=\frac{7}{4}$
Vậy $A_{\min}=\frac{7}{4}$. Giá trị này đạt được khi $x-y=x+\frac{1}{2}=0$

$\Leftrightarrow x=y=\frac{-1}{2}$

Bài 2:

$B=x^2+9y^2+4z^2-2x+12y-4z+20$

$=(x^2-2x+1)+(9y^2+12y+4)+(4z^2-4z+1)+14$

$=(x-1)^2+(3y+2)^2+(2z-1)^2+14$
Vì $(x-1)^2\geq 0; (3y+2)^2\geq 0; (2z-1)^2\geq 0$ với mọi $x,y,z$

$\Rightarrow B\geq 0+0+0+14=14$

Vậy $B_{\min}=14$. Giá trị này đạt được khi $x-1=3y+2=2z-1=0$

$\Leftrightarrow x=1; y=\frac{-2}{3}; z=\frac{1}{2}$

Bài 3:

$C=-x^2-26y^2+10xy-20y-150$
$-C=x^2+26y^2-10xy+20y+150$

$=(x^2+25y^2-10xy)+(y^2+20y+10^2)+50$

$=(x-5y)^2+(y+10)^2+50$
Vì $(x-5y)^2\geq 0; (y+10)^2\geq 0$ với mọi $x,y$

$\Rightarrow -C=(x-5y)^2+(y+10)^2+50\geq 0+0+50=50$

$\Rightarrow C\leq -50$

Vậy $C_{\max}=-50$. Giá trị này đạt được khi $x-5y=y+10=0$

$\Leftrightarrow y=-10; x=-50$

Theo BĐT Cauchy cho 2 số dương, ta có:

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

\(\Rightarrow\frac{x}{2x^2+y^2+5}\le\frac{x}{2\left(xy+x+2\right)}\)(1)

Tương tự ta có: \(\frac{2y}{6y^2+z^2+6}\le\frac{2y}{4\left(yz+y+1\right)}=\frac{y}{2\left(yz+y+1\right)}\)(2)

\(\frac{4z}{3z^2+4x^2+16}\le\frac{4z}{4\left(zx+2z+2\right)}=\frac{z}{zx+2z+2}\)(3)

Cộng theo vế của 3 BĐT (1), (2), (3), ta được: \(\frac{x}{2x^2+y^2+5}+\frac{2y}{6y^2+z^2+6}+\frac{4z}{3z^2+4x^2+16}\)

\(\le\frac{1}{2}\left(\frac{x}{xy+x+2}+\frac{y}{yz+y+1}+\frac{2z}{zx+2z+2}\right)\)

\(=\frac{1}{2}\left(\frac{zx}{xyz+xz+2z}+\frac{xyz}{xyz^2+xyz+xz}+\frac{2z}{zx+2z+2}\right)\)

\(=\frac{1}{2}\left(\frac{zx}{2+xz+2z}+\frac{2}{2z+2+xz}+\frac{2z}{zx+2z+2}\right)\)(Do xyz = 2)

\(=\frac{1}{2}.\frac{zx+2z+2}{zx+2z+2}=\frac{1}{2}\)

Đẳng thức xảy ra khi x = y = 1; z = 2

a. \(x^2+4y^2+z^2=2x+12y-4z-14\)

\(\Leftrightarrow x^2+4y^2+z^2-2x-12y+4z+14=0\)

\(\Leftrightarrow\left(x^2-2x+1\right)+\left(4y^2-12y+9\right)+\left(z^2+4z+4\right)=0\)

\(\Leftrightarrow\left(x-1\right)^2+\left(2y-3\right)^2+\left(z+2\right)^2=0\)

Ta có: \(\left\{{}\begin{matrix}\left(x-1\right)^2\ge0\\\left(2y-3\right)^2\ge0\\\left(z+2\right)\ge0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-1=0\\2y-3=0\\z+2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{3}{2}\\z=-2\end{matrix}\right.\)

b. \(x^2+3y^2+2z^2-2x+12y+4z+15=0\)

\(\Leftrightarrow\left(x^2-2x+1\right)+3\left(y^2+4y+4\right)+2\left(z^2+2z+1\right)=0\)

\(\Leftrightarrow\left(x-1\right)^2+3\left(y+2\right)^2+2\left(z+1\right)^2=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}x-1=0\\y+2=0\\z+1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=-2\\z=-1\end{matrix}\right.\)