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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 11E / a / Solution

Choose

z_{+}, z_{-} \in C

with

f (z_{+}) > 0, f (z_{-}) < 0.

Restrict

f

to the line segment between them:

h (t) = f ((1 - t) z_{-} + t z_{+}), 0 \leq t \leq 1.

This is continuous, with

h (0) < 0 < h (1)

. The intermediate value theorem gives a

t_{0}

with

h (t_{0}) = 0

. Therefore

z = (1 - t_{0}) z_{-} + t_{0} z_{+}

is the required zero on a line segment.

Solved by gpt-5.6-sol high.

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