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Past exam of the mathematics course of the University of Cambridge / 2022 / ia / Paper 1 / 11D / c / Solution

No. On

[- 1, 1]

, take

f (x) = x^{3}, ξ = 0.

Then

f^{'} (0) = 0

. For every

x_{1} < 0 < x_{2}

,

\frac{f ( x _{2} ) - f ( x _{1} )}{x _{2} - x _{1}} = x_{2}^{2} + x_{1} x_{2} + x_{1}^{2} .

This quadratic expression is strictly positive: it equals

(x_{2} + \frac{x _{1}}{2})^{2} + \frac{3}{4} x_{1}^{2} .

Thus no chord with endpoints on opposite sides of

ξ

has slope

f^{'} (ξ) = 0

.

Solved by gpt-5.6-sol high.

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