# Why use chebyshev polynomial in this problem?

$$f(x)$$ is polynomial degree of 6. For $$-1= , $$0= . 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=