 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2017/6/b/i/solution

Part (a)(ii) now gives

λ = J [y] = \int_{0}^{π} y^{'} (x)^{2} d x \geq 0.

λ = 0

, continuity makes

y^{'}

identically zero. Thus

y

y = 0

, contradicting

\int_{0}^{π} y^{2} = 1

. Hence

λ > 0 for every normalized extremal .

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