9. Let $f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e$, for any $x \in[-1,1]$, we have $f(x) \in[-1,1]$, find the maximum value of $|c|$.
Let $g(x)=\frac{f(x)+f(-x)}{2}=a x^{4}+c x^{2}+e$, then for any $x \in[-1,1]$, we have $|g(x)| \leqslant 1$. Thus, $8 \geqslant 4\left|g\left(\frac{\sqrt{2}}{2}\right)\right|+|g(1)|+3|g(0)| \geqslant\left|4\left(\frac{1}{4} a+\frac{1}{2} c+e\right)-(a+c+e)-3 e\right|$ $=|(a+2 c+4 e)-(a+c+e)-3 e|=|c| \Rightarrow|c| \leqslant 8$.
On the other hand, take $f(x)=8 x^{4}-8 x^{2}+1=8\left(x^{2}-\frac{1}{2}\right)^{2}-1$, clearly satisfying for any $x \in[-1,1]$, we have $f(x) \in[-1,1]$. Therefore, the maximum value of $|c|$ is 8.