13. $f(x)$ is a function defined on $[-1,1]$ and is an even function, $g(x)$'s graph is symmetric to the graph of $f(x)$ about the line $x=1$. When $x \in[2,3]$, $g(x)=2 a(x-2)-3(x-2)^{3}$, where $a>\frac{9}{2}$, find: (1) the analytical expression of $f(x)$; (2) the monotonicity of $f(x)$.
13. (1) Let $P(x, y)$ be any point on the graph of $y=f(x)$, and the point $P$ symmetric to the line $x=1$ is $Q(2-x, y)$. When $-1 \leqslant x \leqslant 0$, $2-x \in [2,3]$, then $y=f(x)=g(2-x)=3 x^{3}-2 a x$. Since $f(x)$ is an even function, when $0 < x \leqslant 1$, $f(x)=-3 x^{3}+2 a x$, hence $f(x)=\left\{\begin{array}{ll}3 x^{3}-2 a x, & x \in[-1,0] \\ -3 x^{3}+2 a x, & x \in(0,1] ;\end{array}\right.$
(2) Let $0 < x_{1} < x_{2} \leqslant 1$, then $f\left(x_{1}\right)-f\left(x_{2}\right)=\left(x_{2}-x_{1}\right)\left(3 x_{2}^{2}+3 x_{1} x_{2}+3 x_{1}^{2}-2 a\right)<0$, so $f(x)$ is increasing on $(0,1)$. Similarly, $f(x)$ is decreasing on $[-1,0]$.