Question

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

Answer 1

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