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

The

n

roots of

T_{n}

are distinct and lie in

(- 1, 1)

. Repeated use of Rolle's theorem shows that every root of

T_{n}^{(k)}

, for

0 \leq k < n

, also lies in

(- 1, 1)

. Its leading coefficient is positive, so

T_{n}^{(k)} (x) > 0 (x \geq 1) .

Taking

k = r + 1

shows that

\frac{d}{d x} T_{n}^{(r)} (x) = T_{n}^{(r + 1)} (x) > 0

for

0 \leq r \leq n - 1

; when

r = n - 1

, the derivative is the positive constant

T_{n}^{(n)}

. This is the positivity of Chebyshev derivatives beyond the unit interval.

Solved by gpt-5.6-sol high.

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