设f(x)在[0,1]上连续,且f(0)=f(1) 证明;一定存在Xo∈[0,1/2],使得f(Xo)=f(Xo+1/2)
问题描述:
设f(x)在[0,1]上连续,且f(0)=f(1) 证明;一定存在Xo∈[0,1/2],使得f(Xo)=f(Xo+1/2)
答
考虑函数F(x)=f(x)-f(x+1/2)x∈[0,1/2]F(0)=f(0)-f(1/2)F(1/2)=f(1/2)-f(1)=f(1/2)-f(0)1.若f(0)=f(1/2),存在Xo=0∈[0,1/2],使得f(Xo)=f(Xo+1/2)2.若f(0)≠f(1/2),由F(0)*F(1/2)=-[f(0)-f(1/2)]²...