(1)比较x^6+1与x^4+x^2的大小,其中x∈R

题目

(1)比较x^6+1与x^4+x^2的大小,其中x∈R
(2)若x

答案

（1）
(x^6+1)-(x^4+x^2)
=(x^6-x^4)-(x^2-1)
=x^4(x^2-1)-(x^2-1)
=(x^4-1)(x^2-1)
=(x^2+1)(x^2-1)(x^2-1)
=(x^2+1)(x^2-1)²
≥0
所以x^6+1≥x^4+x^2,当x=±1时它们相等
（2）
(x²+y²)(x-y)-(x²-y²)(x+y)
=(x²+y²)(x-y)-(x-y)(x+y)(x+y)
=(x-y)[(x²+y²)-(x+y)²]
=(x-y)(x²+y²-x²-2xy-y²)
=(x-y)(-2xy)
=2xy(y-x)
因为x0
所以2xy(y-x)>0
即(x²+y²)(x-y)-(x²-y²)(x+y)>0
(x²+y²)(x-y)>(x²-y²)(x+y)