 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-2/22h/b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

Every function with the stated two linear pieces can be written

f (x) = a + b x + c x - \frac{1}{2}

for suitable real constants

a, b, c

. Since

x - \frac{1}{2} = (x - \frac{1}{2})^{2},

Q_{i} (x) = a + b x + c P_{i} ((x - \frac{1}{2})^{2}) .

This is a polynomial. The argument of

P_{i}

lies in

[0, 1/4] \subseteq [0, 1]

, so

∥ Q_{i} - f ∥_{\infty} \leq ∣ c ∣ u \in [0, 1] sup ∣ P_{i} (u) - u ∣ ⟶ 0.

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