Finding the value of $a$ which minimizes the absolute maximum of $f(x)$ 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 $[-1,1]$?


Denote $f(x)=x^{2}$ then $f([-1,0])=f(]0,1])$ and so we need to
minimize $f$ over $[0,1]$.
It is clear that $f([0,1])=[0,1]$ and thus $(f+a)([0,1])=[a,a+1]$.
So we can move $[0,1]$ by $a$ and we want to minimize the absolute
value of numbers in $[a,a+1]$.
It is easy to see (by symmetry) that $a=-0.5$ (and there is a formal
way to get to it by examining which numbers in $[a,a+1]$ are positive)
is the desired $a$. 
After we see this choice, it can be easily proved
it is optimal (any change of $a$ to $a+a'$ you will either increase
the interval to include numbers greater then $0.5$ or smaller then
$-0.5$)
