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Ta có: \(\left(x-y+z\right)^2=x^2-y^2+z^2\)
<=> \(x^2+y^2+z^2-2xy-2yz+2zx=x^2-y^2+z^2\)
<=> \(2y^2-2xy-2yz+2zx=0\)
<=> \(\left(2y^2-2yz\right)-\left(2xy-2xz\right)=0\)
<=>\(2y\left(y-z\right)-2x\left(y-z\right)=0\)
<=>\(2\left(y-x\right)\left(y-z\right)=0\)
<=> \(\left[\begin{array}{nghiempt}y-x=0\\y-z=0\end{array}\right.\)
<=> \(\left[\begin{array}{nghiempt}y=x\\y=z\end{array}\right.\)
Với y=x thì mọi giá trị của z đều thỏa mãn.
Với y=z ta có: \(\left(x-2y\right)^2=x^2\)
<=> \(\left[\begin{array}{nghiempt}x-2y=-x\\x-2y=x\end{array}\right.\)
<=> \(\left[\begin{array}{nghiempt}x=y\\x=-y\end{array}\right.\)
=> x=y=z hoặc -x=y=z.
\(A=\frac{x^2}{\left(x-y\right)\left(x-z\right)}+\frac{y^2}{\left(y-x\right)\left(y-z\right)}+\frac{z^2}{\left(z-x\right)\left(z-y\right)}\)
\(=\frac{x^2}{\left(x-y\right)\left(x-z\right)}-\frac{y^2}{\left(x-y\right)\left(y-z\right)}+\frac{z^2}{\left(x-z\right)\left(y-z\right)}\)
\(=\frac{x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-y\right)}{\left(x-y\right)\left(x-z\right)\left(y-z\right)}\)
\(x^2\left(y-z\right)-y^2\left(x-z\right)+z^2\left(x-y\right)\)
\(=x^2y-x^2z-xy^2+y^2z+z^2\left(x-y\right)\)
\(=xy\left(x-y\right)-z\left(x-y\right)\left(x+y\right)+z^2\left(x-y\right)\)
\(=\left(x-y\right)\left[xy-zx-zy+z^2\right]\)
\(=\left(x-y\right)\left[x\left(y-z\right)-z\left(y-z\right)\right]=\left(x-y\right)\left(x-z\right)\left(y-z\right)\)
Vậy A = 1
\(\left(x-1\right)^2\ge0\Rightarrow x^2-2x+1\ge0\Rightarrow x^2+1\ge2x\)
\(\left(y-2\right)^2\ge0\Rightarrow y^2-4y+4\ge0\Rightarrow y^2+4\ge4y\)
\(\left(z-3\right)^2\ge0\Rightarrow z^2-6z+9\ge0\Rightarrow z^2+9\ge6z\)
Do đó: \(\left(x^2+1\right)\left(y^2+4\right)\left(z^2+9\right)\ge2x.4y.6z=48xyz\)
Dấu "=" xảy ra khI: \(\hept{\begin{cases}x-1=0\\y-2=0\\z-3=0\end{cases}\Rightarrow\hept{\begin{cases}x=1\\y=2\\z=3\end{cases}}}\)
Vậy \(C=\frac{1^3+2^3+3^3}{\left(1+2+3\right)^3}=\frac{6^2}{6^3}=\frac{1}{6}\)
Chúc bạn học tốt.
