Sufficient Conditions for Weak Extremum

Theorem
Let $ J $ be a functional such that:


 * $ \displaystyle J \left [ { y } \right ] = \int_a^b F \left ( { x, y, y' } \right ) \mathrm d x, \quad y \left ( { a } \right ) = A, \quad y \left ( { b } \right ) = B $

Let $ y = y \left ( { x } \right ) $ be an extremum.

Let the strengthened Legendre's condition hold.

Let the strengthened Jacobi's necessary condition hold.

Then the functional $ J $ has a weak extremum for $ y = y \left ( { x } \right ) $.