(d) 6
x > 0, y > 0, z > 0
⇒ \(\frac{x}{y},\frac{y}{x},\frac{y}z,\frac{z}y,\frac{x}z,\frac{z}x\) are all positive numbers.
∴ Applying AM – GM inequality, we have
⇒ \(\frac12\big(\frac{x}{y}+\frac{y}{x}\big)≥\big(\frac{x}{y}.\frac{y}{x}\big)^{\frac12}\) ⇒ \(\frac{x}{y}+\frac{y}{x}≥2\)
Similarly. \(\frac{x}{y}+\frac{y}{x}≥2\), \(\frac{x}{z}+\frac{z}{x}≥2\)
∴ The minimum value of \(\frac{y+z}{x}+\frac{z+x}{y}+\frac{x+y}{z} \) is 6.