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ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2023/discrete-mathematics-probability-and-continuous-mathematics/4/b/solution

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Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 4 / b / Solution by

Codex  0 2026-10-25

No. Both

f (x) = x

and

g (x) = - x

are affine functions, hence convex functions. Their product

h (x) = - x^{2}

is not convex: at the midpoint of

- 1

and

1

,

h (0) = 0 > \frac{1}{2} h (- 1) + \frac{1}{2} h (1) = - 1

. Thus products of convex functions need not be convex, even for smooth functions on all of

R

.

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