1. Let the polynomial $f(x)$ satisfy for any $x \in \mathbf{R}$, $f(x+1)+f(x-1)=2 x^{2}-4 x$, then the minimum value of $f(x)$ is $\qquad$ .
1. -2 .

Since $f(x)$ is a polynomial, then $f(x+1), f(x-1)$ and $f(x)$ have the same degree, so $f(x)$ is a quadratic polynomial. Let $f(x)=a x^{2}+b x+c$, then
$$
f(x+1)+f(x-1)=2 a x^{2}+2 b x+2(a+c),
$$

Therefore,
$$
2 a x^{2}+2 b x+2(a+c)=2 x^{2}-4 x,
$$

Thus $a=1, b=-2, c=-1$. Therefore,
$$
f(x)=x^{2}-2 x-1=(x-1)^{2}-2 \geqslant-2 .
$$