Part CCSIR NET June 2025meeting-the-boundary-conditions-does-not-make-a-function-an-extremal

Meeting the boundary conditions does not make a function an extremal

For $α \geq 0$ , consider the functional $J_{α} [y] = \int_{1}^{2} (y^{'})^{2} / x^{α}$ dx defined for all continuously differentiable functions defined on the interval [1, 2] satisfying the conditions y(1) = 1, y(2) = 2. Then which of the following statements are true?

A. $y (x) = (1/15) (x^{4} + 14)$ is an extremal for $J_{3}$ ✓
B. $y (x) = (1/3) (x^{2} + 2)$ is an extremal for $J_{1}$ ✓
C.y(x) = x is an extremal for $J_{0}$ ✓
D. $y (x) = (1/2) (x^{2} - x + 2)$ is an extremal for $J_{1}$

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 converse assumed.

See pricing

50 are analysed free — try those first.

The trap it tests

Converse assumed

The theorem runs one way. You used it in the other.

Drill statements like this

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

Meeting the boundary conditions does not make a function an extremal

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