Geodesic
Theorems on the existence of an implicit function in the theory of necessary and sufficient conditions for an extremum
Sbornik. Mathematics, Tome 82 (1995) no. 1, pp. 65-85 Cet article a éte moissonné depuis la source Math-Net.Ru

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The implicit function theorem and its generalizations are widely used in the theory of extremum problems. Two such generalizations are given in this paper. As applications of them certain results on necessary and sufficient conditions for an extremum are proved. 
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S. V. Burtsev. Theorems on the existence of an implicit function in the theory of necessary and sufficient conditions for an extremum. Sbornik. Mathematics, Tome 82 (1995) no. 1, pp. 65-85. http://geodesic.mathdoc.fr/item/SM_1995_82_1_a2/

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