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设x、y是实数,且x^2+xy+y^2=3.那么x^2-xy+y^2=S.则S的取值范围是?
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设x、y是实数,且x^2+xy+y^2=3.那么x^2-xy+y^2=S.则S的取值范围是?
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答案和解析
∵x^2+y^2≧2xy
∴x^2+xy+y^2=3≧3xy
xy≦1
∵(x+y)^2=x^2+2xy+y^2≧0
∴3+xy≧0
xy≧-3
∴-2≦-2xy≦6
∵x^2-xy+y^2=3-2xy
1≦3-2xy≦9
∴x^2-xy+y^2的取值范围为【1,9】
∴x^2+xy+y^2=3≧3xy
xy≦1
∵(x+y)^2=x^2+2xy+y^2≧0
∴3+xy≧0
xy≧-3
∴-2≦-2xy≦6
∵x^2-xy+y^2=3-2xy
1≦3-2xy≦9
∴x^2-xy+y^2的取值范围为【1,9】
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