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Part CCSIR NET December 2024free-endpoints-give-the-total-derivative-term-teeth

Free endpoints give the total derivative term teeth

Consider the variational problem minimize J[y]=∫01[(y′)2+2025J[y] = \int_{0}^{1}[(y')^{2} + 2025 yy′]dx, where y(0) and y(1) are free. Then which of the following statements are true?

  1. A.There are infinitely many extremals
  2. B.There are more than one but only finitely many extremals
  3. C.There is no extremal
  4. D.There is a unique extremal✓ 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

Related counterexample: A weak minimum of a functional is a strong minimum

More on this topic

The chapter behind this: Euler–Lagrange, first integrals and null Lagrangians — free to read

From Calculus of Variations › Euler–Lagrange equation and standard functionals

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

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