Extremes of the function - in simple language about complex

To understand what the extremum points of a function are, it is not necessary to know about the presence of the first and second derivatives and to understand their physical meaning. First, you need to understand the following:

• Extremums of the function maximize or, conversely, minimize the value of the function in an arbitrarily small neighborhood;
• At the extremum point there should be no discontinuity of the function.

And now the same, only in simple language. Look at the tip of the rod of the ballpoint pen. If the handle is positioned vertically, the writing end up, then the middle of the ball will be the extremum - the highest point. In this case we speak of a maximum. Now, if you turn the pen with the writing end down, there will already be a minimum of function on the middle of the ball. Using the figure given here, you can submit the listed manipulations for a stationery pencil. So, the extremes of a function are always critical points: its maxima or minima. The adjacent section of the graph can be arbitrarily sharp or smooth, but it must exist on both sides, only in this case the point is an extremum. If the graph is present only on one side, this extremum will not appear even if extremum conditions are satisfied on one side of it. Now we will study the extremums of the function from the scientific point of view. In order for a point to be considered an extreme, it is necessary and sufficient that:

• The first derivative equaled zero or did not exist at the point;
• The first derivative changed its sign at this point.

The condition is treated somewhat differently from the point of view of higher-order derivatives: for a function that is differentiable at a point, it is sufficient that there exist an odd-order derivative that is not equal to zero, provided that all derivatives of the lower order must exist and be zero. This is the simplest interpretation of theorems from textbooks of higher mathematics. But for the most ordinary people it is worthwhile to explain this point by example. The basis is an ordinary parabola. Immediately make a reservation, at the zero point, it has a minimum. Very little math:

• First derivative (X ² ) ^| = 2X, for the zero point 2X = 0;
• Second derivative (2X) ^| = 2, for the zero point z = 2.

In this simple way, the conditions that determine the extrema of the function for both first-order derivatives and higher-order derivatives are illustrated. It can be added to this that the second derivative is precisely the same odd-order derivative, which is not equal to zero, which was mentioned above. When it comes to extremums of a function of two variables, the conditions must be satisfied for both arguments. When the generalization takes place, then private derivatives are used. That is, it is necessary to have an extremum at a point, so that both first-order derivatives are equal to zero, or at least one of them does not exist. For the sufficiency of the presence of an extremum, an expression is considered that is the difference of the product of the second-order derivatives and the square of the mixed second-order derivative of the function. If this expression is greater than zero, then the extremum takes place, and if there is equality to zero, then the question remains open, and more research is needed.