Positivity of Chebyshev derivatives beyond the unit interval

Every root of

T_{n}^{(k)}

lies in

(- 1, 1)

and its leading coefficient is positive. Hence

T_{n}^{(k)} (x) > 0

for

x \geq 1

and

0 \leq k \leq n

.

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 2G / d / Solution

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