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令f(x)=lnx/x, x>0 ,求f(x)之最大值並證明π^e

2007-12-07 10:34:32 · 2 個解答 · 發問者 ? 1 in 科學 數學

2 個解答

1.
f'(x)=(1-lnx)/x²
f'(x)>0=>1-lnx>0=>x0 f'(x)<0=>x>e時f(x)遞減

2. π>e,由1.知f(e)>f(π),即lnπ/π< lne / e
同乘以πe=> e lnπ<πlne=>ln(π^e) < ln(e^π)
故π^e < e^π得證

2007-12-07 15:51:32 補充:
3. 由f(x)之遞增遞減知f(e)=lne / e = 1/e 為最大值

2007-12-07 10:50:20 · answer #1 · answered by mathmanliu 7 · 0 0

好!

2007-12-07 11:56:21 · answer #2 · answered by Regal L 7 · 0 0

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