 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2017/ia/paper-1/10d/a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

f

y_{0}

when for every

ε > 0

there is a

δ > 0

such that

∣ x - y_{0} ∣ < δ ⟹ ∣ f (x) - f (y_{0}) ∣ < ε .

Suppose

f

c

and is differentiable there. For sufficiently small

h > 0

\frac{f ( c + h ) - f ( c )}{h} \geq 0, \frac{f ( c - h ) - f ( c )}{- h} \leq 0.

f^{'} (c)

, so it is both nonnegative and nonpositive. Therefore

f^{'} (c) = 0 .

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