3. (2005 Hunan Province Competition Question) For $x \in \mathbf{R}$, the function $f(x+2)+f(x-2)=f(x)$, then it is a periodic function, and one of its periods is
A. 4
B. 6
C. 8
D. 12
3. D Substitute $x-2$ for $x$ in the equation, then we have $f(x)+f(x-4)=f(x-2)$. Thus, $f(x+2)=-f(x-4)$, which leads to $f(x+6)=-f(x)$. Therefore, $f(x+12)=f(x)$, so the answer is D.
Note that the problem states 12 is the smallest positive period of $f(x)$, which is incorrect. For example, when $f(x)=0$. 12 is just one of its periods.