Vì \(\left(2x-1\right)^{2016}\ge0;\left(3y+6\right)^{2014}\ge0;\left(z-1\right)^{2012}\ge0\)

\(\Rightarrow\left(2x-1\right)^{2016}+\left(3y+6\right)^{2014}+\left(z-1\right)^{2012}\ge0\)

Để \(\left(2x-1\right)^{2016}+\left(3y+6\right)^{2014}+\left(z-1\right)^{2012}=0\)\(\Leftrightarrow\left(2x-1\right)^{2016}=0;\left(3y+6\right)^{2014}=0;\left(z-1\right)^{2012}=0\)

\(\Leftrightarrow2x-1=0;3y+6=0;z-1=0\)

\(\Rightarrow x=\dfrac{1}{2};y=-2;z=1\)

\(\Rightarrow4x+y-3z=4.\dfrac{1}{2}+\left(-2\right)-3.1=2-2-3=-3\)

giúp mk vs nha mina !!!

\(\left(2x-1\right)^{2016}+\left(3y+6\right)^{2014}+\left(z-1\right)^{2012}=0\\ \Rightarrow\left\{{}\begin{matrix}\left(2x-1\right)^{2016}=0\Rightarrow x=\dfrac{1}{2}\\\left(3y+6\right)^{2014}=0\Rightarrow x=-2\\\left(z-1\right)^{2012}=0\Rightarrow x=1\end{matrix}\right.\)

\(\Rightarrow4x+y-3z=\left(4\cdot\dfrac{1}{2}\right)+\left(-2\right)-\left(3\cdot1\right)=2+\left(-2\right)+3=3\)

Ta có: \(x+2y+3x=0\Leftrightarrow x=-\left(2y+3z\right)\)

Lại có: \(2xy+6yz+3xz=0\Leftrightarrow x\left(2y+3z\right)+6yz=0\)

\(\Leftrightarrow-\left(2y+3z\right)\left(2y+3z\right)+6yz=0\Leftrightarrow-\left(2y+3z\right)^2+6yz=0\)

\(\Leftrightarrow\left(2y+3z\right)^2-6yz=0\Leftrightarrow4y^2+12yz+9z^2-6yz=0\)

\(\Leftrightarrow4y^2+6yz+9z^2=0\Leftrightarrow\left(2y+\dfrac{3z}{2}\right)^2+\dfrac{27z^2}{4}=0\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(2y+\dfrac{3z}{2}\right)^2=0\\\dfrac{27z^2}{4}=0\end{matrix}\right.\) \(\Rightarrow y=z=0\Rightarrow x=0\)

\(\Rightarrow S=\dfrac{\left(-1\right)^{2019}-1^{2017}+\left(-1\right)^{2015}}{1^{2018}+2.0^{2016}+0^{2014}+2}=\dfrac{-1-1+-1}{1+0+0+2}=\dfrac{-3}{3}=-1\)

Bài 1 :

a) \(\left(3x-1\right)^2-\left(3x+2\right)\left(3x-2\right)=2014\)

\(\Leftrightarrow9x^2-6x+1-\left(9x^2-4\right)=2014\)

\(\Leftrightarrow-6x=2009\)

\(\Leftrightarrow x=-\dfrac{2009}{6}=-334\dfrac{5}{6}\)

b) \(5x^2+4xy+4y^2+4x+1=0\)

\(\Leftrightarrow\left(x^2+4xy+4y^2\right)+\left(4x^2+4x+1\right)=0\)

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

\(\Leftrightarrow\left\{{}\begin{matrix}x+2y=0\\2x+1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-\dfrac{1}{2}\\y=\dfrac{1}{4}\end{matrix}\right.\)

Bài 2 :

Ta có :

\(D=\left(4x^2-12xy+9y^2\right)-\left(9y^2-4\right)-\left(1-4x+4x^2\right)+12xy-4x\)

\(=4x^2-12xy+9y^2-9y^2+4-1+4x-4x^2+12xy-4x=3\)

Vậy biểu thức D không phụ thuộc vào các biến x,y

\(\left(x^3+1\right)\left(y^3+1\right)\left(z^3+1\right)=\dfrac{81}{64}x^3y^3z^3\)

\(\Leftrightarrow\left(x+1\right)\left(y+1\right)\left(z+1\right)\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)=\dfrac{81}{64}x^2y^2z^2\)

\(\Leftrightarrow3xyz\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)=\dfrac{81}{64}x^3y^3z^3\)

 \(\Rightarrow\left[{}\begin{matrix}xyz=0\\\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)=\dfrac{27}{64}x^2y^2z^2\end{matrix}\right.\)

Nếu \(\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)=\dfrac{27}{64}x^2y^2z^2\) 

Ta có:

\(x^2-x+1=\dfrac{3}{4}x^2+\left(\dfrac{x}{2}-1\right)^2\ge\dfrac{3}{4}x^2\)

Tương tự: \(y^2-y+1\ge\dfrac{3}{4}y^2\) ; \(z^2-z+1\ge\dfrac{3}{4}z^2\)

Do các vế của các BĐT trên đều không âm, nhân vế với vế ta được:

\(\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge\dfrac{27}{64}x^2y^2z^2\)

Đẳng thức xảy ra khi và chỉ khi \(x=y=z=\dfrac{1}{2}\) 

Thế vào  điều kiện \(\left(x+1\right)\left(y+1\right)\left(z+1\right)=3xyz\) ko thỏa mãn (loại)

Vậy \(xyz=0\)