math problem?
2016-01-30 5:07 am
By differentiating the function lnx/x ,or otherwise, prove that if e ≤a<b then a^b>b^a
回答 (1)
2016-01-30 5:32 am
✔ 最佳答案
By differentiating the function (ln x) / x,[(ln x) / x]' = (1/x) / x - (ln x) / x² = (1 - ln x) / x²
If x ≥ e, (1 - ln x) / x² ≤ 0
∴ (ln x) / x is decreasing when x ≥ e
(ln a) / a > (ln b) / b
b ln a > a ln b
ln (a^b) > ln (b^a)
a^b > b^a
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