How to find the extremum?

Before learning how to find extremesfunction, it is necessary to understand what is an extremum. The most common definition of an extremum is that it is the smallest or largest value of a function in mathematics on a certain set of a numerical line or graph. In the place where the minimum is located, there is an extremum of the minimum, and where the maximum is the extremum of the maximum. Also in such a discipline as mathematical analysis, local extremes of function are singled out. Now let's look at how to find extremes.

Extremes in mathematics belong to the most importantcharacteristics of the function, they show its largest and smallest value. The extremums are found mainly at the critical points of the functions found. It is worth noting that it is at the extremum point that the function drastically changes its direction. If we calculate the derivative of the extremum point, then, by definition, it must be zero or completely absent. Thus, in order to learn how to find the extremum of a function, it is necessary to perform two successive tasks:

  • find the derivative for the function to be defined by the job;
  • find the roots of the equation.

The sequence of finding the extremum

  1. Write in writing the function f (x), which is given. Find its derivative of the first order f "(x) .That expression, which is obtained, equate to zero.
  2. Now you have to solve the equation that happened. The resulting solutions u will be the roots of the equation, as well as the critical points of the function being determined.
  3. Now we determine which criticalThe points (maximum or minimum) are the roots found. The next step, after we learned how to find the extremum points of a function, is to find the second derivative of the desired function f (x) .It will be necessary to substitute the values ​​of the critical points found in a particular inequality and then calculate what will happen.If it happens so, that the second derivative is greater than zero at the critical point, then it will be the minimum point, and otherwise it will be the maximum point.
  4. It remains to calculate the value of the initial function innecessary points of the maximum and minimum of the function. To do this, substitute the obtained values ​​in the function and calculate. However, it should be noted that if the critical point is the maximum, then the extremum will be maximum, and if it is a minimum, then it will be minimal by analogy.

The algorithm for finding the extremum

In order to generalize the knowledge gained, we will compile a short algorithm for how to find the extremum points.

So, we have considered how to find the extremum pointsfunction. With the help of simple calculations, as well as knowledge of the finding of derivatives, one can find any extremum and calculate it, as well as graphically designate it. Finding extremums is one of the most important sections of mathematics, both in school and in higher education, so if you learn to correctly define them, then studying will become much easier and more interesting.

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