2017-14 Polynomials of degree at most n

Let $$f(x)\in \mathbb R[x]$$ be a polynomial of degree at most $$n$$ such that $x^2+f(x)^2\le 1$ for all $$-1\le x\le 1$$. Prove that $$\lvert f'(x)\rvert \le 2(n-1)$$ for all $$-1\le x\le 1$$.

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