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2.6.6. Extremes FMP

Lecture



The point M o (x o , y o ) is called the maximum point of the function   2.6.6.  Extremes FMP , if for all points ( x, y ) belonging to a sufficiently small neighborhood of M o (x o , y o ) , the inequality   2.6.6.  Extremes FMP . Function value   2.6.6.  Extremes FMP at the maximum point is called the maximum function.

The point M o (x o , y o ) is called the minimum point of the function   2.6.6.  Extremes FMP , if for all points ( x, y ) belonging to a sufficiently small neighborhood of M o (x o , y o ) , the inequality   2.6.6.  Extremes FMP . Function value   2.6.6.  Extremes FMP at the maximum point is called the minimum of the function.

The points of maximum and minimum of a function are called the extremum points of the function, and the values ​​of the function at these points are called the extremes of the function.

A necessary condition for the existence of an extremum FMP: if a differentiable function   2.6.6.  Extremes FMP reaches an extremum at point M o (x o , y o ) , then its first-order partial derivatives at this point are zero, i.e.   2.6.6.  Extremes FMP .

The points at which the partial derivatives are zero are called stationary points. Not every stationary point is an extremum point.

Sufficient condition for the existence of an extremum of FMF:
let M o (x o , y o ) be the stationary point of the function   2.6.6.  Extremes FMP .
Denote:   2.6.6.  Extremes FMP and make a ratio   2.6.6.  Extremes FMP .

Then:

  • if Δ> 0 , then the value of the function   2.6.6.  Extremes FMP there is an extremum, and this is the maximum if A <0 and the minimum if A > 0 ;
  • if Δ <0 , then the value of the function   2.6.6.  Extremes FMP Extremum is not;
  • if Δ = 0 , then further research is required.

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Mathematical analysis. Differential calculus

Terms: Mathematical analysis. Differential calculus