To understand what is the point of extremum, not necessarily aware of the presence of first and second derivatives and understand their physical meaning.First you need to understand the following:
- extremes maximize function or, conversely, to minimize the value of the function in an arbitrarily small neighborhood;
- at the extremum point there should be discontinuity.
And now the same thing only in simple language.Look at the tip of a pen.If the handle is vertical, writing end up, most middle ball will extremum - the highest point.In this case we speak of the maximum.Now, if you turn the writing end down, into the middle of the ball will be a minimum of a function.With the help of the figures shown here, one can imagine manipulations listed for stationery pencil.So extremes functions - it is always a critical point: its highs or lows.The adjacent portion of the graph can be arbitrarily sharp or smooth, but it must exist on both sides, but in this case the point is the peak.If the schedule is present only on one side, the point of extremum, this will not even be in the case with one side extremum conditions are met.Now we examine the extremes of function from a scientific point of view.In order to qualify as an extremum point, it is necessary and sufficient that:
- first derivative equal to zero or is not there at the point;
- first derivative changes sign at this point.
condition is treated somewhat differently in terms of derivatives of a higher order: for a function differentiable at a point, it is sufficient that there be a derivative of odd order, unequal to zero, despite the fact that all derivatives of a lower order must exist and be equal to zero.This is the most simple interpretation of theorems from textbooks of higher mathematics.But for most ordinary people it is an example to clarify this point.The basis is an ordinary parabola.Outset at zero it has a minimum.Quite a bit of mathematics:
- first derivative (X2) | = 2X, 2X to zero = 0;
- second derivative (2X) | = 2, for zero point 2 = 2.
such simple way illustrate the conditions that determine the extremes functions and first-order, and higher order derivatives.You can add to this that the second derivative is just a derivative of the very odd order, nonzero, mentioned just above.When it comes about the extremes of a function of two variables, the conditions must be met for both arguments.When there is a generalization, then in the course are the partial derivatives.That is, the need for the presence of an extremum at the point that the two first order derivatives equal to zero, or at least one of them did not exist.To investigate the adequacy of having extremum expression representing the difference between the work of the second order and the square of the mixed second-order derivative function.If this expression is greater than zero, then the extremum is the place to be, and if there is equal to zero, then the question remains open, and the need to conduct additional studies.
