x+(4/x-1) (x>1)最小值
来源:百度知道 编辑:UC知道 时间:2024/07/05 08:15:30
求解,希望有步骤..谢谢!
x+4/(x-1)
=(x-1)+4/(x-1)+1
>=2√[(x-1)*4/(x-1)]+1
=5
当且仅当(x-1)=4/(x-1)时,等号成立
此时x=3
x+(4/x-1)
=x-1+4/(x-1)+1
≥1+2√((x-1)*4/(x-1))
=1+4
=5
当且尽当x-1=4/(x-1)
x=3时取5
设f(x)=x+(4/x-1)
f'(x)=1-4/x^2 = 0 所以x=2 或 -2 又因为 x>1
所以x=2
f(x)的最小值=2+4/2-1 =3
x+4/x-1大于等于2*根号四-1=3
最小值为3,当且仅当X=4/X,即x2=4,x=2或x=-2(舍,因为x>1)
即在x=2时取等号
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