设f(x)在[0,1]上连续,在(0,1内可导,且f(1)=0.求证:存在€0,1,使f'(§)=-f(§)/§

问题描述:

设f(x)在[0,1]上连续,在(0,1内可导,且f(1)=0.求证:存在€0,1,使f'(§)=-f(§)/§
RT

设函数g(x)=f(x)*x则g(0)=f(0)*0=0g(1)=f(1)*1=0由于f(x)在[0,1]上连续,在(0,1)内可导,则g(x)在[0,1]上连续,在(0,1)内可导,且g(0)=g(1),由罗尔定理存在§∈(0,1)使g'(§)=0g'(§)=f'(§)§+f(§)=0f'(§)§=-f(§)由...