 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ib/paper-1/17c/a/solution

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

If the bound is to hold for all

f \in C^{3} [- 1, 1]

, the error functional must annihilate every polynomial of degree at most two, because such a polynomial has

f^{'''} = 0

. Applying it to

1, x, x^{2}

gives

a_{- 1} + a_{0} + a_{1} - a_{- 1} + a_{1} a_{- 1} + a_{1} = 0, = 0, = 2.

Therefore

a_{- 1} = 1, a_{0} = - 2, a_{1} = 1,

η (f) = f (- 1) - 2 f (0) + f (1)

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