I know that this is an elementary problem in calculus and so it has a routine way of proof. I faced it and brought it up here just because it is one of R.A.Silverman's interesting problem. Let me learn your approaches like an student. Thanks.
What value of $a$ minimizes the absolute maximum of the function $$f(x)=|x^2+a|$$ in the interval $[-1,1]$?