 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ib/paper-2/7h/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Let

z = t x + (1 - t) y

. Apply the hypothesis with first argument

z

and second argument successively

x, y

f (z) - f (x) \leq (1 - t) λ (z)^{T} (y - x),

f (z) - f (y) \leq t λ (z)^{T} (x - y) .

Multiply these inequalities by

t

and

1 - t

respectively and add. The right sides cancel, leaving

f (z) \leq t f (x) + (1 - t) f (y)

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