 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ia/paper-1/11e/d/solution

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

For

x > 0

[0, x]

f (x) / x = f' (c)

for some

c < x

. Since

f'

is increasing,

f (x) / x \leq f' (x)

. Therefore

(\frac{f ( x )}{x})' = \frac{x f ' ( x ) - f ( x )}{x ^{2}} \geq 0,

and part (c) proves that the quotient is increasing.

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