实数x、y满足x2+y2=4,则x+y-xy的最大值为_.

问题描述:

实数x、y满足x2+y2=4,则x+y-xy的最大值为______.

∵实数x、y满足x2+y2=4,
∴可设x=2cosθ,y=2sinθ.
令t=sinθ+cosθ=

2
sin(θ+
π
4
)(θ∈[0,2π)),
t∈[−
2
2
]

则t2=1+2sinθcosθ,可得2sinθcosθ=t2-1.
∴x+y-xy=2cosθ+2sinθ-4sinθcosθ
=2t-2(t2-1)
=−2(t−
1
2
)2+
5
2
5
2

当且仅当t=
1
2
时,x+y-xy取得最大值为
5
2

故答案为:
5
2