若实数a,b满足ab一4a一b十1=0(a>1),求(a+1)(b+2)的最小值

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若实数a,b满足ab一4a一b十1=0(a>1),求(a+1)(b+2)的最小值若实数a,b满足ab一4a一b十1=0(a>1),求(a+1)(b+2)的最小值若实数a,b满足ab一4a一b十1=0(

若实数a,b满足ab一4a一b十1=0(a>1),求(a+1)(b+2)的最小值
若实数a,b满足ab一4a一b十1=0(a>1),求(a+1)(b+2)的最小值

若实数a,b满足ab一4a一b十1=0(a>1),求(a+1)(b+2)的最小值
∵a>1,∴a-1>0.
令(a+1)(b+2)=k,则:ab+2a+b+2=k,又ab-4a-b+1=0,
∴(ab+2a+b+2)-(ab-4a-b+1)=k,∴6a+2b+1=k,∴b=(k-1-6a)/2.
将b=(k-1-6a)/2代入到ab-4a-b+1=0中,得:
a[(k-1-6a)/2]-4a-(k-1-6a)/2+1=0,
∴a(k-1)-6a^2-8a-(k-1)+6a+2=0,
∴(a-1)(k-1)
=6a^2+2a-2=6[(a-1)+1]^2+2(a-1)=6(a-1)^2+14(a-1)+6,
∴k-1=6(a-1)+6/(a-1)+14≧12+14=26,∴k≧27.
∴k的最小值是27,即:(a+1)(b+2)的最小值是27.