The Weierstrass Necessary Condition
We return now to a consideration of further necessary conditions for a minimizing function x*(·) of the integral J(·). Thus far we have shown the necessity of the Euler condition in its various forms; we recall that it is a necessary condition for both a weak local, and hence a strong local as well as a global minimum and maximum of J(·). Now we shall deduce a condition, the Weierstrass† condition, that is necessary for a strong local, and hence a global minimum of J(·); it is not necessary for a weak local minimum.
KeywordsLocal Minimum Local Maximum Global Minimum Discontinuity Point Excess Function
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