The extreme of the functions of two variables

Definition 11.1: A function has minimum (maximum) at a point if for all points which belong to sufficiently small neighborhood of a point.

Definition 11.2: The points in which function have maximum or minimum are called the points of extreme of the function.

Definition 11.3: The points in which partial derivatives of the first order are not exist or equal to zero are called the critical points or points suspicious on extreme.

Definition 11.4: The points in which all the partial derivatives of the first order exist and are equal to zero are called the stationary points.

Example 11.1. Find critical points of the function

.

Solution: Find partial derivatives

;

.

These derivatives exist for all and . It means that function have stationary points if and .

Compose the system equations

.

Solution of this system will be a stationary point of the given function.

Theorem11.1 (the necessary conditions for extreme). If the function at a point attains extreme, then this point is a critical points of the function.

Theorem11.2 (the sufficient conditions for extreme). Let the function be continuous together with its partial derivatives of the first and second orders and let a point be a stationary point of the function. Denote ,

then = =.

If the function has extreme at a point which is maximum if and minimum if .

If there is no extreme at a point .

If the properties of the second derivatives do not provide any answer to the question of existence of an extreme and further investigation is needed.

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