# (T)Chebyshev Polynomials, again (II)

Calculus Level 5

Let $P_6 [-1, 1] = \{p : [-1,1] \longrightarrow \mathbb{R}; \space \text{ p is a monic polynomial of degree 6 with coefficients in } \mathbb{R}\},$ and if $$p \in P_6 [-1, 1]$$, let's define $|| p ||_{\infty} = \text{ maximum } \{ |p(t)| \text{ such that } \space t\in [-1, 1] \space \}$. Due to Weierstrass extrem value theorem this maximum exits.

Example.-

$$|| p(t) = t^6 ||_{\infty} = 1$$

Find $\displaystyle \min_{p \in P_6 [-1, 1]} || p ||_{\infty}$ ,i.e, Minimize $$|| p ||_{\infty}$$ such that $$p \in P_6 [-1, 1]$$. Give the exact answer.

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