997

记 ${\displaystyle {\displaystyle \underset{x\in [a,b]}{max}\{f(x)\}}}$ 表示 ${\displaystyle f(x)}$ 在区间 ${\displaystyle [a,b]}$ 上的最大值，则 ${\displaystyle {\displaystyle \underset{x\in [0,1]}{max}\{|x^2-x+c|\}}}$ 取得最小值时，${\displaystyle c=}$_____.

记 ${\displaystyle M(c)={\displaystyle \underset{x\in [0,1]}{max}\{|x^2-x+c|\}}}$, 则分别代入 ${\displaystyle x=0,x={\displaystyle \frac{1}{2}}}$, 得$$M(c)\ge |c|,M(c)\ge {\displaystyle |c-\frac{1}{4}|.}$$从而$$\begin{array}{r}2M(c)\ge |c|+|c-\frac{1}{4}|\ge |c-(c-\frac{1}{4})|=\frac{1}{4},\end{array}$$
也即${\displaystyle {\displaystyle M(c)\ge \frac{1}{8}}}$, 取等时${\displaystyle {\displaystyle |c|=|c-\frac{1}{4}|\Rightarrow c=\frac{1}{8}}}$. 充分性不难验证.