Part BCSIR NET December 2024even-modes-vanish-at-the-midpoint

Even modes vanish at the midpoint

Let S denote the set of all solutions of the Euler-Lagrange equation of the variational problem: minimize $J [y] = \int_{0}^{1} (y^{2} + (y^{'})^{2})$ dx, subject to $y (0) = 0, y (1) = 0, \int_{0}^{1} y^{2}$ dx = 1. Then the set { $φ (1/2) : φ \in S$ } is equal to

A.{ $- 2, 2$ }
B.{ $2 / k : k \in Z, k \neq = 0$ }
C.{ $2/ k : k \in N$ }
D.{ $- 2, 0, 2$ }✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

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Even modes vanish at the midpoint

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