Question

若实数x，y满足xy〉0，x^2*y=2，则xy+x^2的最小值是

同题
麻烦写下过程谢谢

Answer 1

xy+x^2
=(1/2)xy+(1/2)xy+x^2
>=3*{[(1/2)xy]*[(1/2)xy]*[x^2]}^(1/3)
=3*(1/4)^(1/3) *(x^4 *y^2)^(1/3)
=3*2^(-2/3) *(x^2 *y)^(2/3)
=3*2^(-2/3) *2^(2/3)
=3

最小值是3

Answer 2

xy>0
x^2*y=2>0
所以 x>0,y>0
xy+x^2=2/x+x^2=1/x+1/x+x^2>=3√(1/x*1/x*x^2)=3
等号当且仅当 x=1,y=2时成立