y=(x²+3x+8)/(x-1)
=[(x-1)(x+4)+12]/(x-1)
=x+4 +12/(x-1)
=(x-1)+12/(x-1)+5
≥2√[(x-1)*12/(x-1)]+5
=2√12+5
=4√3+5
当且仅当 x-1=12/(x-1),即 x=1+2√3时,等号成立
∴ 函数y=x²+3x+8/x-1的最小值 4√3+5