Ta có: \(x^3+y^3=\left(x+y\right)\left(x^2+y^2-xy\right)\ge\left(x+y\right)\left(2xy-xy\right)=xy\left(x+y\right)\)

\(\Rightarrow VT\le\dfrac{1}{xy\left(x+y\right)+xyz}+\dfrac{1}{yz\left(y+z\right)+xyz}+\dfrac{1}{zx\left(z+x\right)+xyz}\)

\(\Rightarrow VT\le\dfrac{1}{x+y+z}\left(\dfrac{1}{xy}+\dfrac{1}{yz}+\dfrac{1}{zx}\right)=\dfrac{1}{x+y+z}.\left(\dfrac{x+y+z}{xyz}\right)=\dfrac{1}{xyz}\) (đpcm)

Dấu "=" xảy ra khi \(x=y=z\)

Do \(x+\dfrac{1}{y}=y+\dfrac{1}{z}=z+\dfrac{1}{x}\)

=> \(\left\{{}\begin{matrix}x+\dfrac{1}{y}=y+\dfrac{1}{z}\Leftrightarrow x-y=\dfrac{1}{z}-\dfrac{1}{y}\Leftrightarrow x-y=\dfrac{y-z}{yz}\\y+\dfrac{1}{z}=z+\dfrac{1}{x}\Leftrightarrow y-z=\dfrac{1}{x}-\dfrac{1}{z}\Leftrightarrow y-z=\dfrac{z-x}{xz}\\z+\dfrac{1}{x}=x+\dfrac{1}{y}\Leftrightarrow z-x=\dfrac{1}{y}-\dfrac{1}{x}\Leftrightarrow z-x=\dfrac{x-y}{xy}\end{matrix}\right.\)

=> \(\left(x-y\right)\left(y-z\right)\left(z-x\right)=\dfrac{\left(y-z\right)\left(z-x\right)\left(x-y\right)}{x^2y^2z^2}\)

<=> \(\left(x-y\right)\left(y-z\right)\left(z-x\right)x^2y^2z^2=\left(y-z\right)\left(z-x\right)\left(x-y\right)\)

<=> \(\left(x-y\right)\left(y-z\right)\left(z-x\right)\left(x^2y^2z^2-1\right)=0\)

=> \(\left(x-y\right)\left(y-z\right)\left(z-x\right)=0\) hoặc \(x^2y^2z^2-1=0\)

=> x=y=z hoặc xyz=1 hoặc xyz=-1

\(x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}\)

\(\Rightarrow\hept{\begin{cases}x-y=\frac{1}{z}-\frac{1}{y}=\frac{y-z}{yz}\\x-z=\frac{1}{x}-\frac{1}{y}=\frac{y-x}{xy}\\y-z=\frac{1}{x}-\frac{1}{z}=\frac{z-x}{xz}\end{cases}}\)

\(\Rightarrow\left(x-y\right)\left(x-z\right)\left(y-z\right)=\frac{\left(y-z\right)\left(y-x\right)\left(z-x\right)}{\left(xyz\right)^2}\)

\(\Rightarrow\left(xyz\right)^2=1\Leftrightarrow\orbr{\begin{cases}xyz=1\\xyz=-1\end{cases}}\).

biet dap an roi can gi phai hoi zo bn ??????????