已知x，y，z＞0，求xy+z+yx+z+zx+y的最小值．

Question

已知x，y，z＞0，求(x)/(y+z)+(y)/(x+z)+(z)/(x+y)的最小值．

Answer 1

∵x，y，z＞0，
∴（x+y+y+z+z+x）((x)/(y+z)+(y)/(x+z)+(z)/(x+y))
≥x+y+z(x(x+y))/(y+z)+(y(x+y))/(x+z)+(y(y+z))/(x+z)+(z(y+z))/(x+y)+(x(z+x))/(y+z)+(z(z+x))/(x+y)
≥x+y+z+2√(xy)+2√(yz)+2√(yz)
=(√(x)+√(y)+√(z))2，
∴(x)/(y+z)+(y)/(x+z)+(z)/(x+y)≥