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Past exam of the mathematics course of the University of Cambridge / 2022 / ib / Paper 1 / 17C / b / Solution

The Peano kernel theorem says that if a continuous linear functional

L

annihilates polynomials of degree below

m

, then

L (f) = \int_{a}^{b} K (t) f^{(m)} (t) d t, K (t) = L (\frac{( x - t ) _{+}^{m - 1}}{( m - 1 )!}) .

Here

m = 3

and

L (f) = f^{''} (0) - f (- 1) + 2 f (0) - f (1) .

Applying

L

to

(x - t)_{+}^{2} /2

gives

K (t) = {\frac{1}{2} (1 + t)^{2}, - \frac{1}{2} (1 - t)^{2}, - 1 \leq t < 0, 0 \leq t \leq 1.

Hence

∣ L (f) ∣ \leq ∥ f^{'''} ∥_{\infty} \int_{- 1}^{1} ∣ K (t) ∣ d t = \frac{1}{3} ∥ f^{'''} ∥_{\infty} .

The smallest candidate is therefore

c = \frac{1}{3} .

Solved by gpt-5.6-sol high.

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