作者JohnMash (Paul)
看板Math
标题Re: [中学] TRML不知哪一年
时间Tue Aug 16 01:43:36 2011
※ 引述《ntucc (整个想完蛋..好累!)》之铭言：
: 1 1
: 1. ______ __ _________ 的最大值？ 当m,n皆为正整数
: m+n+1 (m+1)(n+1)
First, consider m,n are both positive REAL
f(m,n)=mn/(m+n+1)/(m+1)/(n+1)=g(m,n)*n/(n+1)
g(m,n)=m/(m^2+(n+2)m+n+1)
1/g(m,n)=m+n+2+(n+1)/m=n+2+h(m,n)
h(m,n)=m+(n+1)/m
when n is fixed
h(m,n)>=2√(n+1) and minimum occurs at m^2=n+1
That is, max f(m,n) occurs at m^2=n+1 for n fixed
U(m)=f(m,m^2-1)=m(m^2-1)/(m^2+m)/(m+1)/m^2=(m-1)/(m^3+m^2)
U(1)=0, U(2)=1/12,
U(2+k)=(1+k)/(12+16k+7k^2+k^3)<(1+k)/(12+12k)=1/12 when k>0
That is, max f(m,n) CANNOT occur at n>3 for m,n positive real
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we need to check
f(m=1,n=1)=1/12
f(1,2)=1/12
f(2,1)=1/12
f(2,2)=4/45
f(2,3)=1/12
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MAX is f(2,2)=4/45
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※ 编辑: JohnMash 来自: 112.104.89.190 (08/16 11:30)