柯西不等式：(a²+b²)(c²+d²)≥(ac+bd)² 　　等號(hào)成立條件：ad=bc,即a/c=b/d
椐柯西不等式,得
[√(x²+4)+√(y²+9)+√(z²+16)]²
=x²+4+y²+9+z²+16+2√[(x²+4)(y²+9)]+2√[(y²+9)(z²+16)]+2√[(z²+16)(x²+4)]
≥x²+4+y²+9+z²+16+2(xy+6)+2(yz+12)+2(zx+8)
=(x+y+z)²+(2+3+4)²
=442
故當(dāng)且僅當(dāng)x/y=2/3,y/z=3/4,z/x=4/2,
即x=38/9,y=19/3,z=76/9時(shí)等號(hào)成立,
此時(shí)u|min=√442.