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 understand their physical meaning. First you need to understand the following:
- extrema 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 in the function.
And now the same thing, only in plain language. Look at the tip of the ballpoint pen shaft. If the pen is positioned vertically with the writing end up, then the very middle of the ball will be the extremum - the highest point. In this case, they talk about the maximum. Now, if you turn the pen with the writing end down, then at the middle of the ball there will already be a minimum of function. Using the drawing shown here, you can imagine the listed manipulations for a clerical pencil. So, the extrema of a function are always critical points: its maximums or minimums. The adjacent plot 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 on only one side, this point will not be an extremum even if, on one side, the conditions of the extremum are satisfied. Now we study the extrema of the function from a scientific point of view. In order for a point to be considered an extremum, it is necessary and sufficient to:
- the first derivative was zero or did not exist at the point;
- the first derivative changed its sign at this point.
The condition is interpreted somewhat differently from the point of view of higher-order derivatives: for a function differentiable at a point, it is sufficient that there exists an odd-order derivative non-zero, while all lower-order derivatives must exist and be equal to zero. This is the simplest interpretation of theorems from textbooks of higher mathematics. But for the most ordinary people it is worth explaining this point by example. An ordinary parabola is taken as the basis. Immediately make a reservation, at the zero point she has a minimum. Just a little math:
- first derivative (X 2 ) | = 2X, for the zero point 2X = 0;
- second derivative (2X) | = 2, for the zero point 2 = 2.
In such a simple way, the conditions determining the extrema of the function for both first-order derivatives and higher-order derivatives are illustrated. We can add to this that the second derivative is just that very derivative of odd order, non-zero, which was mentioned a little above. When it comes to extrema of the function of two variables, the conditions must be satisfied for both arguments. When generalization occurs, partial derivatives are used. That is, it is necessary for the extremum to exist at a point so that both first-order derivatives are zero, or at least one of them does not exist. For the sufficiency of the presence of an extremum, we study an expression representing the difference of the product of 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 has a place to be, and if there is equality to zero, then the question remains open, and more research is needed.