证明:对任意实数x,y,有x4+y4>=0.5xy(x+y)2
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证明:对任意实数x,y,有x4+y4>=0.5xy(x+y)2
4和2是指四次方和平方
4和2是指四次方和平方
2x^4+2y^4-xy(x+y)^2
= 2x^4+2y^4-xy(x^2+2xy+y^2)
= 2x^4+2y^4-x^3y-2(xy)^2-xy^3
= x^4+y^4-2(xy)^2+x^4+y^4-x^3y-xy^3
= (x^2-y^2)^2+x^3(x-y)-y^3(x-y)
= (x^2-y^2)^2+(x-y)(x^3-y^3)
= (x^2-y^2)^2+(x-y)^2(x^2+xy+y^2)≥0
所以2x^4+2y^4-xy(x+y)^2≥0
2x^4+2y^4≥xy(x+y)^2
x^4+y^4≥0.5xy(x+y)^2
= 2x^4+2y^4-xy(x^2+2xy+y^2)
= 2x^4+2y^4-x^3y-2(xy)^2-xy^3
= x^4+y^4-2(xy)^2+x^4+y^4-x^3y-xy^3
= (x^2-y^2)^2+x^3(x-y)-y^3(x-y)
= (x^2-y^2)^2+(x-y)(x^3-y^3)
= (x^2-y^2)^2+(x-y)^2(x^2+xy+y^2)≥0
所以2x^4+2y^4-xy(x+y)^2≥0
2x^4+2y^4≥xy(x+y)^2
x^4+y^4≥0.5xy(x+y)^2
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