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Part BCSIR NET December 2024a-total-derivative-term-cannot-affect-the-extremal

A total derivative term cannot affect the extremal

If φ∈C4[0,1]\varphi \in C^{4}[0, 1] is the extremal of the variational problem minimize J[y]=∫01(J[y] = \int_{0}^{1}(yy′+(y′′)2)' + (y'')^{2})dx, subject to y(0) = 0, y′(0) = 1, y(1) = 2, y′(1) = 4, then φ(1/2)\varphi(1/2) is equal to

  1. A.5/8✓ Correct
  2. B.3/4
  3. C.3/8
  4. D.5/4

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 hypothesis dropped.

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50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

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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