Question

设函数f(x)是定义在R上的可导偶函数,且图像关于点(1/2,1)对称,则f'(1)+f'(2)+f'(2^2)+…f'

Answer 1

y=f(x),f(-x)=f(x),f'(-x)=-f'(x)=f'(x),y'=f'(x)是奇函数,-f'(0)=f'(-0)=f'(0),f'(0)=0,且图像关于点(1/2,1)对称,所以2-y=f(1-x),y'=f'(1-x),f'(x)=f'(1-x)=-f'(x-1),f'(x+1)=f'(x-1),f'(x+2)=f'(x).

f'(1)=f'(0)=0,f'(2)=f'(2+0)=f'(0)=0,f'(4)=f'(2+2)=f'(2)=f'(0)=0,f'(8)=f'(6+2)=f'(6)=f'(4+2)=f'(4)=0.

f'(2^100)=0

所以原式的值为0