MATH.10:4.5 - Establish the reach of the conclusion

Say which candidates the family reaches and which conclusion the calculation proves.

If every admissible candidate can be represented by a member of the family, and DeltaJ(h)>=0 throughout its allowed domain, the base candidate is a global minimizer. If equality occurs only for the base candidate, it is the unique minimizer. An argument restricted to a neighborhood establishes the corresponding local result.

A narrower family can still supply an improving candidate or a necessary condition. Extending that condition to a stronger conclusion requires coverage of the relevant variations or an applicable sufficiency theorem. A convexity argument can sometimes supply the latter; recover its hypotheses before using it. Repeated failure of a chosen local move supplies no general global-optimality guarantee.

For a physical action, derive the stationarity condition required by the physical model. Establish a minimum only if the functional and allowed histories support that stronger conclusion. The free-history example below does so by a nonnegative finite difference.