MATH.10:4.4 - Derive a condition with the allowed directions

For minimization, an admissible h with DeltaJ(h)<0 supplies an improvement. For a local conclusion, use families whose candidates approach the base candidate as h approaches zero, under the neighborhood notion of the problem.

At a local minimum, a differentiable family allowing sufficiently small changes of both signs must have a=0: a positive or negative a would give a decrease in one of those directions. If only small nonnegative h are allowed, the necessary condition is a>=0. For other parameter domains, test the directions actually available.

When a is zero, inspect the remaining change. The differences h^2, -h^2 and h^3 all have zero derivative at zero but respectively give a minimum, a maximum and neither on a two-sided neighborhood. This distinction changes whether the candidate is selected, rejected or needs a stronger argument.

For a vector of variation parameters, keep their joint constraints. Checking one coordinate at a time can miss an improving combination.