Part CCSIR NET December 2024small-values-permit-large-slopes

Small values permit large slopes

Define S := { $y \in C^{1} [0, 1] : y (0) = y (1) = 0$ }, ‖y‖ $(\infty) := max (x \in [0, 1])$ |y(x)| for every $y \in S, B_{0} (0, ε) :=$ {y ∈ S : ‖y‖ $(\infty) < ε$ }, $B_{1} (0, ε) :=$ {y ∈ S : ‖y‖ $(\infty) +$ ‖y′‖ $(\infty) < ε$ }. Consider the functional $J : S \to R$ given by $J [y] = \int_{0}^{1} [(y^{'})^{2} - 2 x (y^{'})^{4}]$ dx, then there exists an $ε > 0$ such that

A.J[y] ≤ J[0] for every $y \in B_{0} (0, ε)$
B.J[y] ≤ J[0] for every $y \in B_{1} (0, ε)$
C.J[y] ≥ J[0] for every $y \in B_{0} (0, ε)$
D.J[y] ≥ J[0] for every $y \in B_{1} (0, ε)$ ✓

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

Small values permit large slopes

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