Question

f(x+y)+f(x-y)=2f(x)乘f(y) 且f(0)不等于0 f(c/2)=0 求证f(x+2c)=f(x)

Answer 1

y=0

f(x)+f(x)=2f(x)f(0)

f(0)不等于0, f(0)=1.

y=c/2

f(x+c/2)+f(x-c/2)=2f(x)f(c/2)=0

f(x+c/2)=-f(x-c/2)

x=x+c/2

f(x+c)=-f(x)

x=x+c

f(x+2c)=-f(x+c)=f(x)

f(x+2c)=f(x),得证.