Question

（1）设x＜y＜0,试比较(x²+y²)(x-y)与(x²-y²)(x+y)的大小；
（2）已知a,b,c∈{正实数}，且a²+b²=c²,当n∈N,n＞2时比较cⁿ与aⁿ+bⁿ的大小.

Answer 1

（1）(x²+y²)(x-y)＞(x²-y²)(x+y)（2）aⁿ+bⁿ＜cⁿ

（1）方法一 （x²+y²）(x-y)-(x²-y²)(x+y)
=(x-y)［x²+y²-(x+y)²］=-2xy(x-y),
∵x＜y＜0,∴xy＞0,x-y＜0,
∴-2xy(x-y)＞0,
∴(x²+y²)(x-y)＞(x²-y²)(x+y).
方法二 ∵x＜y＜0,∴x-y＜0,x²＞y²,x+y＜0.
∴(x²+y²)(x-y)＜0,(x²-y²)(x+y)＜0,
∴0＜=＜1,
∴(x²+y²)(x-y)＞(x²-y²)(x+y).
(2)∵a,b,c∈{正实数}，∴aⁿ,bⁿ,cⁿ＞0,
而=+.
∵a²+b²=c²,则+=1,
∴0＜＜1,0＜＜1.
∵n∈N,n＞2,
∴＜,＜,
∴=+＜=1,
∴aⁿ+bⁿ＜cⁿ.