(1)
证:a^3+1/a^3-(a^2+1/a^2)
=1/a^3(a^6+1-a^5-a)
=1/a^3(a^5-1)(a-1)
∵a>1
∴1/a^3(a^5-1)(a-1)>0
a^3+1/a^3-(a^2+1/a^2)>0
a^3+1/a^3>a^2+1/a^2>0
(2)若a0
a>0,a^3>0
两边 x a^3 得
a^6+1>a^5+a
a^6-a^5-a+1>0
a^5(a-1)-(a-1)>0
(a^5-1)(a-1)>0
a>1时,a^5-1>0,a-1>0 不等式成立
a0且a≠1时,对任意实数x,y,x>y,x+y>0,证明:a^x + 1/a^x > a^y + 1/a^y.
证:a^x + 1/a^x - (a^y + 1/a^y) = (a^x - a^y)(1- 1/(a^(x+y)))
当00,则a^x + 1/a^x - (a^y + 1/a^y) >0.
所以a^x + 1/a^x > a^y + 1/a^y.