Question

已知实数x大于等于0 y大于等于0 求证 x5+y5>=x3y2+x2y3d

Answer 1

x^5+x^5-x^3y^2-x^2y^3

=x^3(x^2-y^2)-y^3(x^2-y^2)

=(x^2-y^2)(x^3-y^3)

=(x-y)(x+y)(x-y)(x^2+xy+y^2)

=(x-y)^2(x+y)(x^2+xy+y^2)

因x大于等于0 y大于等于0

所以x+y≥0 x^2+xy+y^2≥0

而(x-y)^2≥0

故(x-y)^2(x+y)(x^2+xy+y^2)≥0

所以x5+y5>=x3y2+x2y3

得证