How to find the extremum?
Before learning to find the extremes of a function, it is necessary to understand what an extremum is. The most common definition of extremum is that it is the smallest or largest value of a function used in mathematics on a certain set of a numerical line or graph. In the place where the minimum is located, the extreme of the minimum appears, and where the maximum is the extremum of the maximum. Also in such a discipline as mathematical analysis, local extremes of a function are distinguished. Now let's look at how to find the extremes.
Extremes in mathematics are among the most important characteristics of a function, they show its largest and smallest value. Extremums are found mainly at the critical points of the functions found. It is worth noting that it is at the point of extremum that the function radically changes its direction. If we calculate the derivative of the extremum point, then, according to the definition, it must be zero or it will be completely absent. Thus, to find out how to find the extremum of a function, you must perform two consecutive tasks:
- find the derivative for the function to be determined by the task;
- find the roots of the equation.
The sequence of finding the extremum
- Issue in writing the function f (x) that is specified. Find its first order derivative f '(x). The expression that turns out, equate to zero.
- Now you have to solve the equation that turned out. The resulting solutions will be the roots of the equation, as well as critical points of the defined function.
- Now we determine exactly which critical points (maximum or minimum) the found roots are. The next step, after we learned how to find the extremum points of the function, is to find the second derivative of the desired function f '(x). It will be necessary to substitute the values of the found critical points into a specific inequality and then calculate what happens. If it happens 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.
- It remains to calculate the value of the initial function at the required points of the maximum and minimum of the function.To do this, we substitute the obtained values into the function and calculate. However, it is worth noting that if the critical point turned out to be a maximum, then the extremum will be maximum, and if it is a minimum, then it will be minimum by analogy.
In order to summarize the knowledge gained, let us draw up a short algorithm for finding extremum points.
- We find the domain of definition of a given function and its intervals, which determine exactly at what intervals the function is continuous.
- Find the derivative of the function f '(x).
- We calculate the critical points of the equation y = f (x).
- We analyze changes in the direction of the function f (x), as well as the sign of the derivative f '(x), where the critical points divide the domain of definition of this function.
- Now we determine whether each point on the graph is a maximum or a minimum.
- Find the values of the function at those points that are extremes.
- We fix the result of this study - extremes and gaps of monotony. That's all. Now we looked at how to find the extremum at any interval. If you need to find an extremum on a certain interval of a function, then this is done in a similar way, only the limits of the research being carried out are necessarily taken into account.
So, we looked at how to find the extremum points of a function. With the help of simple calculations, as well as knowledge of finding derivatives, you can find any extremum and calculate it, as well as graphically designate it. Finding extremes is one of the most important branches of mathematics, both at school and at the Higher Educational Institution, therefore, if you learn to identify them correctly, it will become much easier and more interesting to learn.
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