设f(x)=ax^2+bx+c(a≠0),如果对任意x属于[-1,1],都有|f(x)≤1]

日期:2012-01-28 22:38:55 人气:3

设f(x)=ax^2+bx+c(a≠0),如果对任意x属于[-1,1],都有|f(x)≤1]

|f(0)|=|c|<=1,|f(1)|=|a+b+c|<=1,|f(-1)|=|a-b+c|<=1 =>|2a+2c|<=|a+b+c|+|a-b+c|<=2 =>|a+c|<=1 |2a+b|=|(a+b+c)+(a+c)-2c|<=|a+b+c|+|a+c|+2|c|<=4 |2a-b|=|(a-b+c)+(a+c)-2c|<=4 因为x∈[-1,1] 当ab>0时 |2ax+b|<=|2a+b|<=4
    A+
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