 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-1/10f/a/solution

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

[a, b]

m g (x) \leq f (x) g (x) \leq M g (x)

because

g (x) > 0

. To see the implication directly from the definition, every Riemann sum of the nonnegative continuous function

(f - m) g

is nonnegative, so its limit is nonnegative:

\int_{a}^{b} (f - m) g \geq 0.

Applying the same argument to

(M - f) g

gives

\int_{a}^{b} (M - f) g \geq 0

m \int_{a}^{b} g (x) d x \leq \int_{a}^{b} f (x) g (x) d x \leq M \int_{a}^{b} g (x) d x .

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