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5.3. Extremum functions

Lecture



Function value   5.3.  Extremum functions called function maximum   5.3.  Extremum functions , if for any point x from some sufficiently small neighborhood of the point x o the inequality   5.3.  Extremum functions . The point x o is called in this case the maximum point of the function.   5.3.  Extremum functions .

Function value   5.3.  Extremum functions called the minimum function   5.3.  Extremum functions , if for any point x from some sufficiently small neighborhood of the point x o the inequality   5.3.  Extremum functions . The point x o in this case is called the minimum point of the function.   5.3.  Extremum functions .

The maximum or minimum of a function is called a function extremum . The point of maximum or minimum of a function is called the extremum point of the function .

Necessary condition for the existence of an extremum: if a differentiable function   5.3.  Extremum functions reaches an extremum at the point x o , then its first-order derivative at this point is zero, i.e.   5.3.  Extremum functions .

Points where the derivative   5.3.  Extremum functions or does not exist, are called critical points.

Sufficient condition for the existence of an extremum: if x o is a critical point of the function and when passing through it, the derivative   5.3.  Extremum functions changes the sign from plus to minus, then the point x o is the maximum point, and the value of the function   5.3.  Extremum functions - maximum function; if at the transition through the point x o derivative   5.3.  Extremum functions changes the sign from minus to plus, then the point x o is the minimum point, and the value   5.3.  Extremum functions - minimum function; if at the transition through the point x o the derivative of the sign does not change, then there is no extremum at the point, and the value   5.3.  Extremum functions not an extremum of function.


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Mathematical analysis. Differential calculus

Terms: Mathematical analysis. Differential calculus