7.74 For $x \in R$, the function $f(x)$ satisfies $f(x+4)+f(x-4)=$ $f(x)$, then it is a periodic function, and the common minimum period for such functions is
(A) 8.
(B) 12.
(C) 16.
(D) 24.
(E) 32.
(48th American High School Mathematics Examination, 1997)
[Solution] Regarding $x-4$ as $x$ in the given equation, we have:
$$
f(x)+f(x-8)=f(x-4)
$$

Substituting the above equation back into the given equation, we get:
$$
\begin{aligned}
& f(x+4)+f(x)+f(x-8)=f(x) \\
\therefore \quad f(x+4) & =-f(x-8),
\end{aligned}
$$

Substituting $x+8$ for $x$ in (*) gives $\quad f(x)=-f(x+12)$.
Similarly, substituting $x+20$ for $x$ in (*) gives:
$$
f(x+24)=-f(x+12) .
$$

In summary, $\quad f(x)=-f(x+12)=f(x+24)$.
If $f(x)$ has a smaller period $T$, it should be a divisor of 24, but substituting it into the given equation does not work. Therefore, the smallest period is 24.
Hence, the answer is $(D)$.