Chọn phương án D.

\(\begin{aligned}
S&=\,\dfrac{x+y}{z}+\dfrac{y+z}{x}+\dfrac{z+x}{y}\\
&=\,\dfrac{x}{z}+\dfrac{y}{z}+\dfrac{y}{x}+\dfrac{z}{x}+\dfrac{z}{y}+\dfrac{x}{y}\\
&=\,\left(\dfrac{x}{z}+\dfrac{z}{x}\right)+\left(\dfrac{y}{z}+\dfrac{z}{y}\right)+\left(\dfrac{y}{x}+\dfrac{x}{y}\right).
\end{aligned}\)

Áp dụng bất đẳng thức Cauchy ta có

• \(\dfrac{x}{z}+\dfrac{z}{x}\geq2\)
• \(\dfrac{y}{z}+\dfrac{z}{y}\geq2\)
• \(\dfrac{y}{x}+\dfrac{x}{y}\geq2\)

Từ đó suy ra \(S\geq6\).