Question

已知：a>0,b>0.求证：(a+b)(a^4+b^4)< =2(a^5+b^5)

Answer 1

2(a^5+b^5)-(a+b)(a^4+b^4)

=a^5+b^5-b*a^4-a*b^4

=a^4*(a-b)+b^4*(b-a)

=(a^4-b^4)(a-b)

=(a+b)(a-b)^2(a^2+b^2)

因为a>0;b>0,

所以(a+b)(a-b)^2(a^2+b^2)>=0

即证