实数x,y满足x^2+y^2=4,则x+y-xy的最大值是?

实数x,y满足x^2+y^2=4,则x+y-xy的最大值是?
实数x,y满足x^2+y^2=4,则x+y-xy的最大值是?

实数x,y满足x^2+y^2=4,则x+y-xy的最大值是?
设a = x+y,b = xy,则a²-2b = x²+y² = 4,即b = a²/2-2.
所求式x+y-xy = a-b = -a²/2+a+2 = -(a-1)²/2+5/2 ≤ 5/2.
上述不等式当且仅当a = 1时成立等号.
而不难验证方程组x+y = 1,x²+y² = 4有实数解,故等号能够成立.
因此x+y-xy的最大值就是5/2.