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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $P$ be a real [[polynomial function|MAM9H3SY3Y]] of [[degree|0DPF6WRZW6]] at most $n$.\nWrite $\\norm{Q} = \\max_{-1 \\le x \\le 1} \\abs{Q(x)}$ for a function $Q$ continuous on $[-1, 1]$.\n\n1. $\\norm{P'} \\le n^2 \\norm{P}$.\n2. More generally, for $1 \\le k \\le n$, the $k$-th [[derivative|084NGSTWM7]] satisfies\n$$\\norm{P^{(k)}} \\le \\frac{n^2 (n^2 - 1^2)(n^2 - 2^2) \\cdots (n^2 - (k-1)^2)}{1 \\cdot 3 \\cdot 5 \\cdots (2k - 1)} \\, \\norm{P}.$$\n\n## Remarks\n\nBoth bounds are attained by the [[Chebyshev polynomial|M6EAR1JHPP]] $T_n$, defined by $T_n(\\cos \\theta) = \\cos(n \\theta)$.\nThe constant in the second bound equals $T_n^{(k)}(1)$; for $k = 1$ it is $n^2$.\n\nExample: $P(x) = T_2(x) = 2x^2 - 1$ has $\\norm{P} = 1$ and $P'(x) = 4x$, so $\\norm{P'} = 4 = 2^2 \\norm{P}$.\n\nIn the interior, [[Bernstein's inequality for polynomials|F0YGC89S9R]] gives the stronger pointwise bound $\\abs{P'(x)} \\le \\frac{n}{\\sqrt{1 - x^2}} \\norm{P}$ for $-1 \u003c x \u003c 1$.\nThe factor $n^2$ in the first bound is caused by the endpoints $\\pm 1$, where this pointwise bound degenerates.\n\n## Explanations\n\n### L1 human\n\nA polynomial of degree $n$ that stays between $-1$ and $1$ on the interval $[-1, 1]$ cannot be too steep there.\nIts slope is at most $n^2$ in absolute value.\nThe Chebyshev polynomials, which oscillate as much as possible, are the steepest ones.\n\n### L2 human\n\nMarkov-type inequalities bound the derivative of a polynomial by its size, with a constant depending only on the degree.\nThey are used in approximation theory to prove inverse theorems: if a function is well approximated by polynomials, then it is smooth.\nThe first inequality is due to A. A. Markov and the general one to his brother V. A. Markov.\n",
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