<s>[INST] How to find the maximum value of $|a_{2}|$? 
(Tenth grade high school math competition):Let $$f(x)=a_{0}+a_{1}x+a_{2}x^2+a_{3}x^3+a_{4}x^4$$
  where $a_{i}\in \mathbb R$, and such that for $x\in[-1,1]$ we have
  $$|f(x)|\le 1$$
  Find the maximum of the value $|a_{2}|$.

I know Chebyshev polynomial $T_{4}(x)=8x^4-8x^2+1$ verifies such condition.
But how to prove that it generates the looked for maximum ?
 [/INST] Note that we only have to consider $f$ which are even, since if $f$ is a quartic with $|f(x)|\le 1$ for $-1\le x\le 1$, then $e(x) = \frac{f(x)+f(-x)}{2}$ is also a quartic with $|e(x)|\le 1$ for $-1\le x\le 1$, and $e$ is even and has the same coefficient of $x^2$ as $f$.
As such, let $f(x) = a_0+a_2x^2+a_4x^4$. Then $f(x) = g(x^2)$, where $g(x) = a_0+a_2x+a_4x^2$, and $g$ has the property that $|g(x)|\le 1$ for $0\le x\le 1$ (since $-1\le x\le 1\iff 0\le x^2\le 1$). In particular, $|g(x)|\le 1$ for $x=0$, $\frac{1}{2}$, and $1$. Now
$$ a_2 = 4\cdot\left(a_0+\frac{a_2}{2} + \frac{a_4}{4}\right) - (a_0+a_2+a_4) -3a_0 = 4g\left(\frac{1}{2}\right) - g(1) - 3g(0) $$
and so $g\left(\frac{1}{2}\right)\le 1$, $g(1)\ge -1$, and $g(0)\ge -1$ imply that
$$ a_2 \le 4\cdot 1 - (-1) - 3\cdot(-1) = 8.$$
Similarly, $g\left(\frac{1}{2}\right)\ge 1$, $g(1)\le -1$, and $g(0)\le -1$ imply that $ a_2 \ge -8$. Hence, we must have $|a_2|\le 8$.
Conversely, $g(x) = 8\left(x-\frac{1}{2}\right)^2-1 = 8x^2-8x+1$ satisfies $|g(x)|\le 1$ for $0\le x\le 1$, and hence $f(x) = 8x^4-8x^2+1$ satisfies $|f(x)|\le 1$ for $-1\le x\le 1$. Hence, $8$ is the maximum possible value of $|a_2|$.
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