Question

设x为正实数,n属于N+,求证x^n+x^-n≥x^n-1+x^1-n

Answer 1

建议使用规范表达：x^{n-1}表示x的n-1次方,x^n-1表示x的n次方减一.

Without loss of generality,we can suppose that x>=1 (Otherwise consider 1/x instead of x).The desired inquality can be transformed to

x^{n-1}(x-1)>=x^{-n}(x-1),

i.e.,

x^{n-1}>=x^{-n}.

In fact,x^{n-1}>=1>=1/x>=x^{-n}.This completes the proof.