Part CCSIR NET December 2023weak-vs-strong-extremum

Weak vs strong extremum

Define S = { $y \in C^{1} [0, π] : y (0) = y (π) = 0$ }, ‖f‖ $_{\infty} = max$ |f|, $B_{0} (0, ε) =$ {f ∈ S : ‖f‖ $_{\infty} < ε$ } and $B_{1} (0, ε) =$ {f ∈ S : ‖f‖ $_{\infty} +$ ‖f′‖ $_{\infty} < ε$ }. For the functional $J [y] = \int_{0}^{π} (1 - (y^{'})^{2}) y^{2}$ dx, there exists $ε > 0$ such that

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

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