实数xy满足x+y-2≥0,x+3y-6≤0,x-2≤0则x^2+y^2+2x+2y的最小值

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实数xy满足x+y-2≥0,x+3y-6≤0,x-2≤0则x^2+y^2+2x+2y的最小值实数xy满足x+y-2≥0,x+3y-6≤0,x-2≤0则x^2+y^2+2x+2y的最小值实数xy满足x+

实数xy满足x+y-2≥0,x+3y-6≤0,x-2≤0则x^2+y^2+2x+2y的最小值
实数xy满足x+y-2≥0,x+3y-6≤0,x-2≤0则x^2+y^2+2x+2y的最小值

实数xy满足x+y-2≥0,x+3y-6≤0,x-2≤0则x^2+y^2+2x+2y的最小值
证明x、y大于等于0
∵x+y-2≥0,x+3y-6≤0
∴2-y≤x≤6-3y,2-y≤6-3y,y≤2
∵x-2≤0,x+y-2≥0
∴2-y≤x≤2,y≥0
∵0≤y≤2,x+y-2≥0
∴0≤x≤2
证明x+y与xy的取值范围
∵x+y-2≥0,x+y≥2
又∵x+y≥2√xy
∴2√xy≤2,xy≤1
x²+y²+2x+2y=(x+y)²+2(x+y)-2xy
当x=y=1时,x+y最小,xy最大,原式最小值=4+4-2=6

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