已知x,y都是正实数,求证:(x^2+y^2)^1/2>(x^3+y^3)^1/3
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(x^2+y^2)^(1/2)
(x^3+y^3)^(1/3)
两个式子个六次方
(x^2+y^2)^3-(x^3+y^3)^2
=x^6+3x^4y^2+3x^2y^4+y^6-x^6-2x^3y^3-y^6
=3x^4y^2+3x^2y^4-2x^3y^3
=x^2y^2(3x^2-2xy+3y^2)
=x^2y^2[3(x-y/3)^2+8y^2/3]>=0
若要等于0
则3(x-y/3)^2=0,8y^2/3=0
则x=y=0,和x>0,y>0矛盾
所以(x^2+y^2)^3-(x^3+y^3)^2>0
(x^2+y^2)^3>(x^3+y^3)^2
(x^2+y^2)^(1/2)>(x^3+y^3)^(1/3)
(x^3+y^3)^(1/3)
两个式子个六次方
(x^2+y^2)^3-(x^3+y^3)^2
=x^6+3x^4y^2+3x^2y^4+y^6-x^6-2x^3y^3-y^6
=3x^4y^2+3x^2y^4-2x^3y^3
=x^2y^2(3x^2-2xy+3y^2)
=x^2y^2[3(x-y/3)^2+8y^2/3]>=0
若要等于0
则3(x-y/3)^2=0,8y^2/3=0
则x=y=0,和x>0,y>0矛盾
所以(x^2+y^2)^3-(x^3+y^3)^2>0
(x^2+y^2)^3>(x^3+y^3)^2
(x^2+y^2)^(1/2)>(x^3+y^3)^(1/3)
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