Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 323 / 3 / ii / c

Table of contents
- Solution c

Solution

For

t > 0

, positive homogeneity and concavity give

f (X + t Y) = (1 + t) f (\frac{X}{1 + t} + \frac{t Y}{1 + t}) \geq (1 + t) (\frac{1}{1 + t} f (X) + \frac{t}{1 + t} f (Y)) = f (X) + t f (Y) .

After subtracting

f (X)

, dividing by

t

, and taking the one-sided directional derivative at zero,

\frac{d}{d t}_{t = 0^{+}} f (X + t Y) \geq f (Y) .

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