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4x2 + 2y2 + 2z2-4xy - 4xz+2yz-6y-10z+34=0
<=>(-2x+y+z)2+(y-3)2+(z-5)2=0
<=>\(\left\{{}\begin{matrix}-2x+y+z=0\\y=3\\z=5\end{matrix}\right.\)
<=>\(\left\{{}\begin{matrix}x=4\\y=3\\z=5\end{matrix}\right.\)
Vậy A=\(\left(4-4\right)^{22}+\left(3-4\right)^6+\left(5-4\right)^{2013}=0^{22}+\left(-1\right)^6+1^{2013}=0+1+1=2\)
đề bài sai nhé, 6x phảy là 6y
\(\Leftrightarrow\left(4x^2+y^2+z^2-4xy-4xz+2yz\right)+\left(y^2-6y+9\right)+\left(z^2-10z+25\right)=0\)
\(\Leftrightarrow\left(-2x+y+z\right)^2+\left(y-3\right)^2+\left(z-5\right)^2=0\)
Vì \(\left(-2x+y+z\right)^2\ge0\)
\(\left(y-3\right)^2\ge0\)
\(\left(z-5\right)^2\ge0\)
\(\Rightarrow\left(-2x+y+z\right)^2+\left(y-3\right)^2+\left(z-5\right)^2\ge0\)
Dấu "=" xảy ra \(\Leftrightarrow y=3;z=5;x=4\)
\(\left(x-4\right)^{2015}+\left(y-4\right)^{2015}+\left(z-4\right)^{2015}=\left(4-4\right)^{2015}+\left(3-4\right)^{2015}+\left(5-4\right)^{2015}=0\)
Lời giải:
\(4x^2+2y^2+2z^2-4xy-4xz+2yz-6y-10z+34=0\)
\(\Leftrightarrow (4x^2-4xy+y^2)+y^2+2z^2-2z(2x-y)-6y-10z+34=0\)
\(\Leftrightarrow (2x-y)^2-2z(2x-y)+z^2+y^2+z^2-6y-10z+34=0\)
\(\Leftrightarrow (2x-y-z)^2+(y^2-6y+9)+(z^2-10z+25)=0\)
\(\Leftrightarrow (2x-y-z)^2+(y-3)^2+(z-5)^2=0\)
Do \((2x-y-z)^2; (y-3)^2; (z-5)^2\geq 0, \forall x,y,z\), nên để tổng của chúng bẳng $0$ thì:
\((2x-y-z)^2=(y-3)^2=(z-5)^2=0\Rightarrow \left\{\begin{matrix}
y=3\\
z=5\\
x=4\end{matrix}\right.\)
\(\Rightarrow S=(x-4)^{2014}+(y-4)^{2015}+(z-4)^{2016}=0+(-1)^{2015}+1^{2016}=-1+1=0\)
Ta có : \(4x^2+2y^2+2z^2-4xy+2yz-6y-10z+34=0\)
\(\Leftrightarrow\left(4x^2+y^2+z^2-4xy-4xz+2yz\right)+\left(y^2-6y+9\right)+\left(z^2-10z+25\right)=0\)
\(\Leftrightarrow\left(y+z-2x\right)^2+\left(y-3\right)^2+\left(z-5\right)^2=0\)
\(\Leftrightarrow\hept{\begin{cases}y+z-2x=0\\y=3\\z=5\end{cases}}\Leftrightarrow\hept{\begin{cases}x=4\\y=3\\z=5\end{cases}}\)
Suy ra \(M=2\)
Ta có : 4x^2+2y^2+2z^2-4xy+2yz-6y-10z+34=04x2+2y2+2z2−4xy+2yz−6y−10z+34=0
\Leftrightarrow\left(4x^2+y^2+z^2-4xy-4xz+2yz\right)+\left(y^2-6y+9\right)+\left(z^2-10z+25\right)=0⇔(4x2+y2+z2−4xy−4xz+2yz)+(y2−6y+9)+(z2−10z+25)=0
\Leftrightarrow\left(y+z-2x\right)^2+\left(y-3\right)^2+\left(z-5\right)^2=0⇔(y+z−2x)2+(y−3)2+(z−5)2=0
\(\Leftrightarrow\hept{\begin{cases}y+z-2x=0\\y=3\\z=5\end{cases}}\Leftrightarrow\hept{\begin{cases}x=4\\y=3\\z=5\end{cases}}\)
Suy ra M=2M=2
4x2 + 2y2 + 2z2 - 4xy + 2yz - 4xz - 6y - 10z + 34 = 0
<=> [ ( 4x2 - 4xy + y2 ) - 4xz + 2yz + z2 ] + ( y2 - 6y + 9 ) + ( z2 - 10z + 25 ) = 0
<=> [ ( 2x - y )2 - 2( 2x - y )z + z2 ] + ( y - 3 )2 + ( z - 5 )2 = 0
<=> ( 2x - y - z )2 + ( y - 3 )2 + ( z - 5 )2 = 0
\(\hept{\begin{cases}\left(2x-y-z\right)^2\\\left(y-3\right)^2\\\left(z-5\right)^2\end{cases}}\ge0\forall x,y,z\Rightarrow\left(2x-y-z\right)+\left(y-3\right)^2+\left(z-5\right)^2\ge0\forall x,y,z\)
Đẳng thức xảy ra <=> \(\hept{\begin{cases}2x-y-z=0\\y-3=0\\z-5=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=4\\y=3\\z=5\end{cases}}\)
Thế vào S ta được :
S = ( x - 4 )2020 + ( y - 3 )2020 + ( z - 5 )2020
= ( 4 - 4 )2020 + ( 3 - 3 )2020 + ( 5 - 5 )2020
= 0 + 0 + 0
= 0