Part CCSIR NET December 2024weak-minimum-need-not-be-strong

Weak minimum need not be strong

Define S := { $y \in C^{1} [- 1, 1] : y (- 1) = - 1, y (1) = 3$ }. Let $φ$ be the extremal of the functional $J : S \to R$ given by $J [y] = \int$ ₋ $_{1}^{1} [(y^{'})^{3} + (y^{'})^{2}]$ dx. Define ‖y‖ $(\infty) := max (x \in [- 1, 1])$ |y(x)| for every y ∈ S and let $B_{0} (φ, ε) :=$ {y ∈ S : ‖ $y - φ$ ‖ $(\infty) < ε$ }, $B_{1} (φ, ε) :=$ {y ∈ S : ‖ $y - φ$ ‖ $(\infty) +$ ‖ $y^{'} - φ^{'}$ ‖ $(\infty) < ε$ }. Then which of the following statements are true?

A. $φ (x) = 2 x + 1$ for every x ∈ [−1, 1]✓
B.There exists $ε > 0$ such that $J [y] \geq J [φ]$ for every $y \in B_{0} (φ, ε)$
C.There exists $ε > 0$ such that $J [y] \geq J [φ]$ for every $y \in B_{1} (φ, ε)$ ✓
D.There exists $ε > 0$ such that $J [y] \leq J [φ]$ for every $y \in B_{1} (φ, ε)$

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.

See pricing

50 are analysed free — try those first.

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

Weak minimum need not be strong

More on this topic