Question

已知xy都是正实数且满足4x²+4xy+y²+2x+y-6=0则x（1-y）的最小值

Answer 1

4x²+4xy+y²+2x+y-6=0

(2x+y)²+(2x+y)-6=0

(2x+y+3)(2x+y-2)=0

2x+y+3=0或2x+y-2=0

y=-2x-3或y=2-2x

x(1-y)=x(1+2x+3)

=2x²+4x

=2(x²+2x+1)-2

=2(x+1)²-2

≥-2

∴当x=-1,y=-1时则x（1-y）的最小值是-2

x(1-y)=x(1+2x-2)

=2x²-x

=2(x²-½x+¼)-½

=2(x-½﹚²-½

≥-½

∴当x=½,y=1时则x（1-y）的最小值是-½