Theorem of Monge

The theorem of Monge is used to study the behavior of a Fonction of several variables on the level of a not criticizes $(x\_0,\; y\_0)$:

One poses:

One calculates $\backslash \; Delta\; =\; S\; ^\; \{2\}\; -\; RT$.

Three cases then are distinguished:

• If one has a local extrémum then:
• if $R\; >\; 0$ it is a maximum;
• if it is a minimum;
• If $\backslash \; Delta\; >\; 0$ one has then a Point saddle - or not collar;
• If $\backslash \; Delta\; =\; 0$ it is then necessary to make the traditional study of the not criticizes.

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