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Part CCSIR NET December 2024the-corner-slopes-cap-the-reachable-endpoint

The corner slopes cap the reachable endpoint

For any b∈Rb \in \mathbb{R}, let S(b) denote the set of all broken extremals with one corner of the variational problem: minimize J[y]=∫01((y′)4−3(y′)2)J[y] = \int_{0}^{1}((y')^{4} - 3(y')^{2})dx, subject to y(0) = 0, y(1) = b. Then which of the following statements are true?

  1. A.S(2) has exactly two elements
  2. B.S(1/2) has exactly one element
  3. C.S(2) is empty✓ Correct
  4. D.S(1/2) has exactly two elements✓ 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.

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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

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