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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]=∫01(y2+(y′)2)J[y] = \int_{0}^{1}(y^{2} + (y')^{2})dx, subject to y(0)=0,y(1)=0,∫01y2y(0) = 0, y(1) = 0, \int_{0}^{1}y^{2}dx = 1. Then the set {φ(1/2):φ∈S\varphi(1/2) : \varphi \in S} is equal to

  1. A.{−2,2-\sqrt{2}, \sqrt{2}}
  2. B.{2/k:k∈Z,k≠0\sqrt{2}/k : k \in \mathbb{Z}, k \ne 0}
  3. C.{2/k:k∈N\sqrt{2/k} : k \in \mathbb{N}}
  4. D.{−2,0,2-\sqrt{2}, 0, \sqrt{2}}✓ Correct

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, and why this one tests boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

More on this topic

The chapter behind this: Constrained variational problems — free to read

From Calculus of Variations › Isoperimetric problems

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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