9. Let $f(x)=x^{2}+p x+q, p 、 q \in \mathbf{R}$. If the maximum value of $|f(x)|$ on $[-1,1]$ is $M$, find the minimum value of $M$.

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9. Let $M=\max _{-1 \leq x \leq 1}|f(x)|$, then we have $M \geqslant|f(1)|=|1+p+q|, M \geqslant|f(-1)|=|1-p+q|$, $M \geqslant|f(0)|=|q|$, thus $4 M \geqslant|1+p+q|+2|-q|+|1-p+q| \geqslant |(1+p+q)+2(-q)+(1-p+q)|=2$, hence $M \geqslant \frac{1}{2}$. Example: when $f(x)=x^{2}-\frac{1}{2}$, $\max _{-1 \leq x \leq 1}|f(x)|=\frac{1}{2}$, so the minimum value of $M$ is $\frac{1}{2}$.