高手帮忙做个微分中值定理的题

来源:百度知道 编辑:UC知道 时间:2024/05/15 12:12:32
f(0)=0,f(x)的2阶导数<0,证明对任意x1,x2>0,有f(x1+x2)<f(x1)+f(x2)

use mean value theorem
f(x2)=f(x2)-f(0)=f'(a)*x2 0<a<x2
f(x1+x2)-f(x1)=f'(b)*x2 x1<b<x1+x2
we might assume x2<x1 as well
hence a<x2<x1<b
because f''(x)<0 hence f'(a)>f'(b) x2>0
hence f'(a)*x2>f'(b)*x2
so f(x2)=f(x2)-f(0)=f'(a)*x2>f'(b)*x2=f(x2+x1)-f(x1)
so f(x2)+f(x1)>f(x1+x2)

这个题目好像看过,现在这种题目忘的差不多了,不好意思。到图书馆找吧