At how many points is the derivative of a function positive? At what point is the value of the derivative the largest?

Showing the relationship of the sign of the derivative with the nature of the monotonicity of the function.

Please be extremely careful in the following. Look, the schedule of WHAT is given to you! Function or its derivative

Given a graph of the derivative, then we are only interested in function signs and zeros. No "knolls" and "hollows" are of interest to us in principle!

Task 1.

The figure shows a graph of a function defined on an interval. Determine the number of integer points where the derivative of the function is negative.

Decision:

In the figure, the areas of decreasing function are highlighted in color:

4 integer values ​​fall into these areas of decreasing function.

Task 2.

The figure shows a graph of a function defined on an interval. Find the number of points where the tangent to the graph of the function is parallel or coincident with the line.

Decision:

Since the tangent to the function graph is parallel (or coincides) with a straight line (or, which is the same, ) having slope , zero, then the tangent has a slope .

This in turn means that the tangent is parallel to the axis, since the slope is the tangent of the angle of inclination of the tangent to the axis.

Therefore, we find extremum points on the graph (maximum and minimum points), - it is in them that the functions tangent to the graph will be parallel to the axis.

There are 4 such points.

Task 3.

The figure shows a graph of the derivative of a function defined on the interval . Find the number of points where the tangent to the graph of the function is parallel or coincident with the line.

Decision:

Since the tangent to the graph of the function is parallel (or coincides) with a straight line, which has a slope, then the tangent has a slope.

This in turn means that at the points of contact.

Therefore, we look at how many points on the graph have an ordinate equal to .

As you can see, there are four such points.

Task 4.

The figure shows a graph of a function defined on an interval. Find the number of points where the derivative of the function is 0.

Decision:

The derivative is zero at the extremum points. We have 4 of them:

Task 5.

The figure shows a function graph and eleven points on the x-axis:. At how many of these points is the derivative of the function negative?

Decision:

On intervals of decreasing function, its derivative takes negative values. And the function decreases at points. There are 4 such points.

Task 6.

The figure shows a graph of a function defined on an interval. Find the sum of the extremum points of the function .

Decision:

extremum points are the maximum points (-3, -1, 1) and the minimum points (-2, 0, 3).

The sum of extreme points: -3-1+1-2+0+3=-2.

Task 7.

The figure shows a graph of the derivative of a function defined on the interval . Find the intervals of increasing function . In your answer, indicate the sum of integer points included in these intervals.

Decision:

The figure highlights the intervals on which the derivative of the function is non-negative.

There are no integer points on the small interval of increase, on the interval of increase there are four integer values: , , and .

Their sum:

Task 8.

The figure shows a graph of the derivative of a function defined on the interval . Find the intervals of increasing function . In your answer, write the length of the largest of them.

Decision:

In the figure, all the intervals on which the derivative is positive are highlighted, which means that the function itself increases on these intervals.

The length of the largest of them is 6.

Task 9.

The figure shows a graph of the derivative of a function defined on the interval . At what point on the segment does highest value.

Decision:

We look at how the graph behaves on the segment, namely, we are interested in derivative sign only .

The sign of the derivative on is minus, since the graph on this segment is below the axis.

The derivative of a function is one of difficult topics in the school curriculum. Not every graduate will answer the question of what a derivative is.

This article simply and clearly explains what a derivative is and why it is needed.. We will not now strive for mathematical rigor of presentation. The most important thing is to understand the meaning.

Let's remember the definition:

The derivative is the rate of change of the function.

The figure shows graphs of three functions. Which one do you think grows the fastest?

The answer is obvious - the third. She has the most high speed changes, that is, the largest derivative.

Here is another example.

Kostya, Grisha and Matvey got jobs at the same time. Let's see how their income changed during the year:

You can see everything on the chart right away, right? Kostya's income has more than doubled in six months. And Grisha's income also increased, but just a little bit. And Matthew's income decreased to zero. The starting conditions are the same, but the rate of change of the function, i.e. derivative, - different. As for Matvey, the derivative of his income is generally negative.

Intuitively, we can easily estimate the rate of change of a function. But how do we do it?

What we are really looking at is how steeply the graph of the function goes up (or down). In other words, how fast y changes with x. Obviously, the same function at different points can have different meaning derivative - that is, it can change faster or slower.

The derivative of a function is denoted by .

Let's show how to find using the graph.

A graph of some function is drawn. Take a point on it with an abscissa. Draw a tangent to the graph of the function at this point. We want to evaluate how steeply the graph of the function goes up. A handy value for this is tangent of the slope of the tangent.

The derivative of a function at a point is equal to the tangent of the slope of the tangent drawn to the graph of the function at that point.

Please note - as the angle of inclination of the tangent, we take the angle between the tangent and the positive direction of the axis.

Sometimes students ask what is the tangent to the graph of a function. This is a straight line, which has the only common point with a graph, and as shown in our figure. It looks like a tangent to a circle.

Let's find . We remember that the tangent of an acute angle in right triangle equal to the ratio of the opposite leg to the adjacent one. From triangle:

We found the derivative using the graph without even knowing the formula of the function. Such tasks are often found in the exam in mathematics under the number.

There is another important correlation. Recall that the straight line is given by the equation

The quantity in this equation is called slope of a straight line. It is equal to the tangent of the angle of inclination of the straight line to the axis.

.

We get that

Let's remember this formula. She expresses geometric meaning derivative.

The derivative of a function at a point is equal to the slope of the tangent drawn to the graph of the function at that point.

In other words, the derivative is equal to the tangent of the slope of the tangent.

We have already said that the same function can have different derivatives at different points. Let's see how the derivative is related to the behavior of the function.

Let's draw a graph of some function. Let this function increase in some areas, decrease in others, and with different speed. And let this function have maximum and minimum points.

At a point, the function is increasing. The tangent to the graph drawn at the point forms sharp corner; with positive axis direction. So the derivative is positive at the point.

At the point, our function is decreasing. The tangent at this point forms an obtuse angle; with positive axis direction. Since the tangent of an obtuse angle is negative, the derivative at the point is negative.

Here's what happens:

If a function is increasing, its derivative is positive.

If it decreases, its derivative is negative.

And what will happen at the maximum and minimum points? We see that at (maximum point) and (minimum point) the tangent is horizontal. Therefore, the tangent of the slope of the tangent at these points is zero, and the derivative is also zero.

The point is the maximum point. At this point, the increase of the function is replaced by a decrease. Consequently, the sign of the derivative changes at the point from "plus" to "minus".

At the point - the minimum point - the derivative is also equal to zero, but its sign changes from "minus" to "plus".

Conclusion: with the help of the derivative, you can find out everything that interests us about the behavior of the function.

If the derivative is positive, then the function is increasing.

If the derivative is negative, then the function is decreasing.

At the maximum point, the derivative is zero and changes sign from plus to minus.

At the minimum point, the derivative is also zero and changes sign from minus to plus.

We write these findings in the form of a table:

Let's make two small clarifications. You will need one of them when solving the problem. Another - in the first year, with a more serious study of functions and derivatives.

A case is possible when the derivative of a function at some point is equal to zero, but the function has neither a maximum nor a minimum at this point. This so-called :

At a point, the tangent to the graph is horizontal and the derivative is zero. However, before the point the function increased - and after the point it continues to increase. The sign of the derivative does not change - it has remained positive as it was.

It also happens that at the point of maximum or minimum, the derivative does not exist. On the graph, this corresponds to a sharp break, when it is impossible to draw a tangent at a given point.

But how to find the derivative if the function is given not by a graph, but by a formula? In this case, it applies

Hello! Let's hit the approaching USE with high-quality systematic training, and perseverance in grinding the granite of science !!! ATAt the end of the post there is a competitive task, be the first! In one of the articles in this section, you and I, in which the graph of the function was given, and set various questions concerning extremums, intervals of increase (decrease) and others.

In this article, we will consider the tasks included in the USE in mathematics, in which the graph of the derivative of a function is given, and the following questions are posed:

1. At what point of a given segment does the function take on the largest (or smallest) value.

2. Find the number of maximum (or minimum) points of the function that belong to a given segment.

3. Find the number of extremum points of the function that belong to a given segment.

4. Find the extremum point of the function that belongs to the given segment.

5. Find intervals of increase (or decrease) of the function and in the answer indicate the sum of integer points included in these intervals.

6. Find intervals of increase (or decrease) of the function. In your answer, indicate the length of the largest of these intervals.

7. Find the number of points where the tangent to the graph of the function is parallel to the straight line y = kx + b or coincides with it.

8. Find the abscissa of the point at which the tangent to the graph of the function is parallel to the abscissa axis or coincides with it.

There may be other questions, but they will not cause you any difficulties if you understand and (links are provided to articles that provide the information necessary for solving, I recommend repeating).

Basic information (briefly):

1. The derivative on increasing intervals has a positive sign.

If the derivative at a certain point from some interval has positive value, then the graph of the function on this interval increases.

2. On the intervals of decreasing, the derivative has a negative sign.

If the derivative at a certain point from some interval has negative meaning, then the graph of the function decreases on this interval.

3. The derivative at the point x is equal to the slope of the tangent drawn to the graph of the function at the same point.

4. At the points of extremum (maximum-minimum) of the function, the derivative is equal to zero. The tangent to the graph of the function at this point is parallel to the x-axis.

This needs to be clearly understood and remembered!!!

The graph of the derivative "confuses" many people. Some inadvertently take it for the graph of the function itself. Therefore, in such buildings, where you see that a graph is given, immediately focus your attention in the condition on what is given: a graph of a function or a graph of a derivative of a function?

If it is a graph of the derivative of a function, then treat it like a "reflection" of the function itself, which simply gives you information about this function.

Consider the task:

The figure shows a graph y=f'(X)- derivative function f(X), defined on the interval (–2;21).

We will answer the following questions:

1. At what point of the segment is the function f(X) takes on the largest value.

On a given segment, the derivative of the function is negative, which means that the function decreases on this segment (it decreases from the left boundary of the interval to the right). Thus, the maximum value of the function is reached on the left boundary of the segment, i.e., at point 7.

Answer: 7

2. At what point of the segment is the function f(X)

From this graph of the derivative, we can say the following. On a given segment, the derivative of the function is positive, which means that the function increases on this segment (it increases from the left border of the interval to the right one). Thus, smallest value The function is reached on the left boundary of the segment, that is, at the point x = 3.

Answer: 3

3. Find the number of maximum points of the function f(X)

The maximum points correspond to the points where the sign of the derivative changes from positive to negative. Consider where the sign changes in this way.

On the segment (3;6) the derivative is positive, on the segment (6;16) it is negative.

On the segment (16;18) the derivative is positive, on the segment (18;20) it is negative.

Thus, on a given segment, the function has two maximum points x = 6 and x = 18.

Answer: 2

4. Find the number of minimum points of the function f(X) belonging to the segment .

The minimum points correspond to the points where the sign of the derivative changes from negative to positive. We have a negative derivative on the interval (0; 3), and positive on the interval (3; 4).

Thus, on the segment, the function has only one minimum point x = 3.

*Be careful when writing the answer - the number of points is recorded, not the x value, such a mistake can be made due to inattention.

Answer: 1

5. Find the number of extremum points of the function f(X) belonging to the segment .

Please note that you need to find amount extremum points (these are both maximum and minimum points).

The extremum points correspond to the points where the sign of the derivative changes (from positive to negative or vice versa). On the graph given in the condition, these are the zeros of the function. The derivative vanishes at points 3, 6, 16, 18.

Thus, the function has 4 extremum points on the segment.

Answer: 4

6. Find the intervals of increasing function f(X)

Intervals of increase of this function f(X) correspond to the intervals on which its derivative is positive, that is, the intervals (3;6) and (16;18). Please note that the boundaries of the interval are not included in it (round brackets - boundaries are not included in the interval, square brackets are included). These intervals contain integer points 4, 5, 17. Their sum is: 4 + 5 + 17 = 26

Answer: 26

7. Find the intervals of decreasing function f(X) at a given interval. In your answer, indicate the sum of integer points included in these intervals.

Function Decreasing Intervals f(X) correspond to intervals on which the derivative of the function is negative. In this problem, these are the intervals (–2;3), (6;16), (18;21).

These intervals contain the following integer points: -1, 0, 1, 2, 7, 8, 9, 10, 11, 12, 13, 14, 15, 19, 20. Their sum is:

(–1) + 0 + 1 + 2 + 7 + 8 + 9 + 10 +

11 + 12 + 13 + 14 + 15 + 19 + 20 = 140

Answer: 140

*Pay attention in the condition: whether the boundaries are included in the interval or not. If the boundaries are included, then these boundaries must also be taken into account in the intervals considered in the solution process.

8. Find the intervals of increasing function f(X)

Function increase intervals f(X) correspond to the intervals on which the derivative of the function is positive. We have already indicated them: (3;6) and (16;18). The largest of them is the interval (3;6), its length is 3.

Answer: 3

9. Find the intervals of decreasing function f(X). In your answer, write the length of the largest of them.

Function Decreasing Intervals f(X) correspond to intervals on which the derivative of the function is negative. We have already indicated them, these are the intervals (–2; 3), (6; 16), (18; 21), their lengths are respectively equal to 5, 10, 3.

The length of the largest is 10.

Answer: 10

10. Find the number of points where the tangent to the graph of the function f(X) parallel to the line y \u003d 2x + 3 or coincides with it.

The value of the derivative at the point of contact is equal to the slope of the tangent. Since the tangent is parallel to the straight line y \u003d 2x + 3 or coincides with it, then their slopes are equal to 2. Therefore, it is necessary to find the number of points at which y (x 0) \u003d 2. Geometrically, this corresponds to the number of intersection points of the derivative graph with the straight line y = 2. There are 4 such points on this interval.

Answer: 4

11. Find the extremum point of the function f(X) belonging to the segment .

An extremum point of a function is a point at which its derivative is equal to zero, and in the vicinity of this point, the derivative changes sign (from positive to negative or vice versa). On the segment, the graph of the derivative crosses the x-axis, the derivative changes sign from negative to positive. Therefore, the point x = 3 is an extremum point.

Answer: 3

12. Find the abscissas of the points where the tangents to the graph y \u003d f (x) are parallel to the abscissa axis or coincide with it. In your answer, indicate the largest of them.

The tangent to the graph y \u003d f (x) can be parallel to the x-axis or coincide with it, only at points where the derivative is zero (these can be extremum points or stationary points, in the vicinity of which the derivative does not change its sign). This graph shows that the derivative is zero at points 3, 6, 16,18. The largest is 18.

The argument can be structured like this:

The value of the derivative at the point of contact is equal to the slope of the tangent. Since the tangent is parallel or coincident with the x-axis, its slope is 0 (indeed, the tangent of an angle of zero degrees is zero). Therefore, we are looking for a point at which the slope is equal to zero, which means that the derivative is equal to zero. The derivative is equal to zero at the point where its graph crosses the x-axis, and these are points 3, 6, 16,18.

Answer: 18

The figure shows a graph y=f'(X)- derivative function f(X) defined on the interval (–8;4). At what point of the segment [–7;–3] is the function f(X) takes the smallest value.

The figure shows a graph y=f'(X)- derivative function f(X), defined on the interval (–7;14). Find the number of maximum points of a function f(X) belonging to the segment [–6;9].

The figure shows a graph y=f'(X)- derivative function f(X) defined on the interval (–18;6). Find the number of minimum points of a function f(X) belonging to the interval [–13;1].

The figure shows a graph y=f'(X)- derivative function f(X), defined on the interval (–11; –11). Find the number of extremum points of a function f(X), belonging to the segment [–10; -ten].

The figure shows a graph y=f'(X)- derivative function f(X) defined on the interval (–7;4). Find the intervals of increasing function f(X). In your answer, indicate the sum of integer points included in these intervals.

The figure shows a graph y=f'(X)- derivative function f(X), defined on the interval (–5; 7). Find the intervals of decreasing function f(X). In your answer, indicate the sum of integer points included in these intervals.

The figure shows a graph y=f'(X)- derivative function f(X) defined on the interval (–11;3). Find the intervals of increasing function f(X). In your answer, write the length of the largest of them.

F The figure shows a graph

The condition of the problem is the same (which we considered). Find the sum of three numbers:

1. The sum of the squares of the extrema of the function f (x).

2. The difference of the squares of the sum of the maximum points and the sum of the minimum points of the function f (x).

3. The number of tangents to f (x) parallel to the straight line y \u003d -3x + 5.

The first one to give the correct answer will receive an incentive prize - 150 rubles. Write your answers in the comments. If this is your first comment on the blog, then it will not appear immediately, a little later (do not worry, the time of writing a comment is recorded).

Good luck to you!

Sincerely, Alexander Krutitsikh.

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