(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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