Question

已知a∈R+,比较(a+1)(a^4+1)与(a^2+1)(a^3+1)的大小

Answer 1

(a+1)(a^4+1)－(a^2+1)(a^3+1)＝a^5+a^4+a+1-a^5-a^3-a^2-1=a^4-a^3-a^2+a=a^3(a-1)-a(a-1)

=a(a-1)(a^2-1)=a(a-1)^2(a+1)

∵a∈R+ ∴a＞0 (a-1)^2≥0 a+1＞0

∴(a+1)(a^4+1)－(a^2+1)(a^3+1)≥0 即 (a+1)(a^4+1)≥(a^2+1)(a^3+1)

∵当(a+1)(a^4+1)－(a^2+1)(a^3+1)=a(a-1)^2(a+1)=0时,a=1

∴当且仅当a=1时等号成立