\(x^2+4y^2+z^2+14\ge2x+12y+4z\)
\(\left(x^2-2x+1\right)+\left(4y^2-12y+9\right)+\left(z^2-4z+4\right)\ge0\)
\(\left(x-1\right)^2+\left(2y-3\right)^2+\left(z-2\right)^2\ge0\) (luôn đúng)
Dấu bằng xảy ra \(\Leftrightarrow\left\{{}\begin{matrix}\left(x-1\right)^2=0\\\left(2y-3\right)^2=0\\\left(z-2\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x-1=0\\2y-3=0\\z-2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=1\\y=\dfrac{3}{2}\\z=2\end{matrix}\right.\)
Ta có: điều phải chứng minh
`<=> x^2 - 2x + 1 + 4y^2 - 12y + 9 + z^2 - 4z + 4 >=0`
`<=> (x-1)^2 + (2y-3)^2 + (z-2)^2 >=0`
Dấu bằng xảy ra `<=> {(x=1), (y=3/2), (z=2):}`
