f(x)=ax^2+bx+c在0,1上满足-1≤f(x)≤1,试求|a|+|b|+|c|的最大值

Question

f(x)=ax^2+bx+c在【0,1】上满足-1≤f(x)≤1,试求|a|+|b|+|c|的最大值

Answer 1

f(1)=a+b+c;
f(0)=c;
f(1/2)=a/4+b/2+c;
得a=-4*f(1/2)+2*f(0)+2*f(1),b=4*f(1/2)-f(1)-3*f(0),c=f(0);
因此|a|+|b|+|c|=|-4*f(1/2)+2*f(0)+2*f(1)|+|4*f(1/2)-f(1)-3*f(0)|+|f(0)|<=|4*f(1/2)|+|2*f(0)|+|2*f(1)|+|4*f(1/2)|+|f(1)|+|3*f(0)|+|f(0)|<=4+2+2+4+1+3+1=17;当且仅当f(1)=f(0)=-f(1/2)=-1(或1）时成立,对应a=-8,b=8,c=-1(或a=8,b=-8,c=1);