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$f(x)$ is polynomial degree of 6. For $-1=<x=<1$ , $0=<f(x)=<1$ . What is maximum value of leading coefficient of $f(x)$.

I saw solution, solution claim $g(x)=2f(x) -1$ and $g(x)=T_6 (x)$ so answer is 16. But why it use chebyshev polynomial? I can't catch the idea. I solve it by using leading coefficient of $f(x)$ is equivalent to leading coefficient of $h(x)$ that polynomial degree of 3 and for 0=

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