Find the polynomial $f(x)$ which have the following property Find the polynomial $p(x)=x^2+px+q$ for which $\max\{\:|p(x)|\::\:x\in[-1,1]\:\}$ is minimal.
This is the 2nd exercise from a test I gave, and I didn't know how to resolve it. Any good explanations will be appreciated. Thanks!

An answer that uses more advanced mathematics (which is probably not what you want) ...
The monic polynomial of degree $n$ that has minimal sup-norm on $[-1,1]$ is $T_n(x)/2^{n-1}$, where $T_n(x)$ is the Chebyshev polynomial of degree $n$. In fact, this minimum norm property can serve as the defining property of Chebyshev polynomials.
In our case, $n=2$, so the desired polynomial is $T_2(x)/2$ = $(2x^2 - 1)/2 = x^2 - \tfrac12$.
More about Chebyshev polynomials here.
