f(x)=lnx-1/x,x≥2,证明:f(x-1)≤2x-5,我算到h(x)=ln(x+1)-1/(x+1)-2x+5≤h(2)=ln3...
f(x)=lnx-1/x,x≥2,证明:f(x-1)≤2x-5,我算到h(x)=ln(x+1)-1/(x+1)-2x+5≤h(2)=ln3+2/3后就算不下去了
人气:110 ℃ 时间:2020-06-04 07:51:27
解答
证:f(x-1)=ln(x-1)-1/(x-1)
f(x-1)-2x+5=ln(x-1)-1/(x-1)-2(x-1)+3=lnt-1/t-2t+3(t=x+1,t≥1)
设g(t)=lnt-1/t-2t+3(t≥1),g'(t)=1/t+1/t^2-2在t≥1时恒小于0,g(t)单调递减
g(t)≤g(t)max=g(1)=0
即f(x-1)≤2x-5
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