已知x不等于y,x^4+y^4与x^3y+xy^3的大小关系是?

Question

已知x不等于y,x^4+y^4与x^3y+xy^3的大小关系是?

Answer 1

(x^4+y^4)-(x^3·y+x·y^3)
=(x^3-y^3)(x-y)
=(x²+xy+y²)(x-y)²,
若xy≥0(x≠y),
则(x²+xy+y²)(x-y)²﹥0,
∴x^4+y^4﹥x^3·y+x·y^3；
若xy＜0(x≠y),
则x²+xy+y²=(x+y)²-xy＞0,
∴(x²+xy+y²)(x-y)²﹥0,
∴x^4+y^4﹥x^3·y+x·y^3；
综上x^4+y^4﹥x^3·y+x·y^3.