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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 $[-1,1]$?

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Is the answer $a=-0.5$ ? – Belgi Jan 5 '13 at 16:38
Ok I'll post my answer :) – Belgi Jan 5 '13 at 16:38
up vote 2 down vote accepted

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 $-0.5$)

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I feel that changes from a to a+a'. Yes I see – Babak S. Jan 5 '13 at 16:47
@BabakSorouh - I imagined moving $[0,1]$ around and found $a=0.5$, I didn't really do anything smart here. – Belgi Jan 5 '13 at 16:49
Thanks for the time you spent. @Andre, Belgi and Cameron. Thanks for sharing your ideas. – Babak S. Jan 5 '13 at 16:53

Note the symmetry of $x^2+a$ about the $y$-axis. Given the interval, the absolute maximum is $\max\{-a,1+a\}$. (Why?) Minimization, then, will be achieved when $-a=1+a$--that is, when $a=-\frac12$.

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About to write exactly that. – Per Alexandersson Jan 5 '13 at 16:36

In my opinion, a calculus student would likely start by looking at the graph of this function for a few values of $a$. Then, they would notice that the absolute maxima occur on the endpoints or at $x=0$. So, looking at $f(-1)=f(1)=|1+a|$ and $f(0)=|a|$, I think it would be fairly easy for a calculus student to see that $a=-\frac{1}{2} \hspace{1 mm} $ minimizes the absolute maximum.

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Yes, you're right. I think the way which all of you suggested is applicable. Thanks. – Babak S. Jan 5 '13 at 17:08

Let $y=x^2$. We want to minimize the maximum of $|y+a|$ as $y$ ranges over $[0,1]$. This happens at $a=-1/2$.

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We have $0\leq x^2\leq1$ for the values given. We want to shift this function so that its minimum negative value is equal to its maximum positive value - cutting this range of values in half. So we have to shift it down by 0.5.

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