893

$f(x)$ 定义域为 $\mathbb{R}$, $f\left({\displaystyle \frac{1}{2}}\right)\ne 0$. 若 $f(x+y)+f(x)f(y)=4xy$, 则
A. $f(-\frac{1}{2})=0$
B. $f\left(\frac{1}{2}\right)=-2$
C. $f(x-\frac{1}{2})$ 是偶函数
D. $f(x+\frac{1}{2})$ 是减函数

首先，带入 ${\displaystyle x=\frac{1}{2},y=0}$:$$\begin{array}{r}f\left(\frac{1}{2}\right)(1+f(0))=0.\end{array}$$
而${\displaystyle f\left(\frac{1}{2}\right)\ne 0}$，故$f(0)=-1$.然后，带入${\displaystyle x=\frac{1}{2},y=-\frac{1}{2}}$:$$\begin{array}{r}f\left(\frac{1}{2}\right)f(-\frac{1}{2})=0\Rightarrow f(-\frac{1}{2})=0.\end{array}$$
然后，带入$x=x+{\displaystyle \frac{1}{2},y=-\frac{1}{2}}$:$$\begin{array}{r}f(x)=-2x-1,\end{array}$$
因此容易判断ABD正确，C错误.