Today we invite you to explore and plot a function graph with us. After a careful study of this article, you will not have to sweat for a long time to complete this kind of task. It is not easy to explore and build a graph of a function, the work is voluminous, requiring maximum attention and accuracy of calculations. To facilitate the perception of the material, we will gradually study the same function, explain all our actions and calculations. Welcome to the amazing and fascinating world of mathematics! Go!

Domain

In order to explore and plot a function, you need to know a few definitions. A function is one of the basic (basic) concepts in mathematics. It reflects the dependence between several variables (two, three or more) with changes. The function also shows the dependence of sets.

Imagine that we have two variables that have a certain range of change. So, y is a function of x, provided that each value of the second variable corresponds to one value of the second. In this case, the variable y is dependent, and it is called a function. It is customary to say that the variables x and y are in For greater clarity of this dependence, a graph of the function is built. What is a function graph? This is a set of points on the coordinate plane, where each value of x corresponds to one value of y. Graphs can be different - a straight line, hyperbola, parabola, sinusoid and so on.

A function graph cannot be plotted without exploration. Today we will learn how to conduct research and plot a function graph. It is very important to make notes during the study. So it will be much easier to cope with the task. The most convenient study plan:

1. Domain.
2. Continuity.
3. Even or odd.
4. Periodicity.
5. Asymptotes.
6. Zeros.
7. Constancy.
8. Ascending and descending.
9. Extremes.
10. Convexity and concavity.

Let's start with the first point. Let's find the domain of definition, that is, on what intervals our function exists: y \u003d 1/3 (x ^ 3-14x ^ 2 + 49x-36). In our case, the function exists for any values ​​of x, that is, the domain of definition is R. This can be written as xОR.

Continuity

Now we are going to explore the discontinuity function. In mathematics, the term "continuity" appeared as a result of the study of the laws of motion. What is infinite? Space, time, some dependencies (an example is the dependence of the variables S and t in motion problems), the temperature of the heated object (water, frying pan, thermometer, and so on), a continuous line (that is, one that can be drawn without taking it off the sheet pencil).

A graph is considered continuous if it does not break at some point. One of the most good examples such a graph is a sine wave, which you can see in the picture in this section. The function is continuous at some point x0 if a number of conditions are met:

• a function is defined at a given point;
• the right and left limits at a point are equal;
• the limit is equal to the value of the function at the point x0.

If at least one condition is not met, the function is said to break. And the points at which the function breaks are called break points. An example of a function that will “break” when displayed graphically is: y=(x+4)/(x-3). Moreover, y does not exist at the point x = 3 (since it is impossible to divide by zero).

In the function that we are studying (y \u003d 1/3 (x ^ 3-14x ^ 2 + 49x-36)) everything turned out to be simple, since the graph will be continuous.

Even, odd

Now examine the function for parity. Let's start with a little theory. An even function is a function that satisfies the condition f (-x) = f (x) for any value of the variable x (from the range of values). Examples are:

• module x (the graph looks like a jackdaw, the bisector of the first and second quarters of the graph);
• x squared (parabola);
• cosine x (cosine wave).

Note that all of these graphs are symmetrical when viewed with respect to the ordinate (i.e. y).

What then is called an odd function? These are those functions that satisfy the condition: f (-x) \u003d - f (x) for any value of the variable x. Examples:

• hyperbola;
• cubic parabola;
• sinusoid;
• tangent and so on.

Please note that these functions are symmetric about the point (0:0), that is, the origin. Based on what was said in this section of the article, an even and odd function must have the property: x belongs to the definition set and -x too.

Let us examine the function for parity. We can see that she doesn't fit any of the descriptions. Therefore, our function is neither even nor odd.

Asymptotes

Let's start with a definition. An asymptote is a curve that is as close as possible to the graph, that is, the distance from some point tends to zero. There are three types of asymptotes:

• vertical, that is, parallel to the y axis;
• horizontal, i.e. parallel to the x-axis;
• oblique.

As for the first type, these lines should be looked for at some points:

• gap;
• ends of the domain.

In our case, the function is continuous, and the domain of definition is R. Therefore, there are no vertical asymptotes.

The graph of a function has a horizontal asymptote, which meets the following requirement: if x tends to infinity or minus infinity, and the limit is equal to a certain number (for example, a). AT this case y=a is the horizontal asymptote. There are no horizontal asymptotes in the function we are studying.

An oblique asymptote only exists if two conditions are met:

• lim(f(x))/x=k;
• lim f(x)-kx=b.

Then it can be found by the formula: y=kx+b. Again, in our case there are no oblique asymptotes.

Function zeros

The next step is to examine the graph of the function for zeros. It is also very important to note that the task associated with finding the zeros of a function occurs not only in the study and plotting of a function graph, but also as an independent task, and as a way to solve inequalities. You may be required to find the zeros of a function on a graph or use mathematical notation.

Finding these values ​​will help you plot the function more accurately. If to speak plain language, then the zero of the function is the value of the variable x, at which y=0. If you are looking for zeros of a function on a graph, then you should pay attention to the points where the graph intersects with the x-axis.

To find the zeros of the function, you need to solve the following equation: y=1/3(x^3-14x^2+49x-36)=0. After doing the necessary calculations, we get the following answer:

sign constancy

The next step in the study and construction of a function (graphics) is finding intervals of sign constancy. This means that we must determine on which intervals the function takes a positive value, and on which intervals it takes a negative value. The zeros of the functions found in the previous section will help us to do this. So, we need to draw a straight line (separate from the graph) and in right order distribute the zeros of the function over it from smallest to largest. Now you need to determine which of the resulting intervals has a “+” sign, and which one has a “-”.

In our case, the function takes a positive value on the intervals:

• from 1 to 4;
• from 9 to infinity.

Negative meaning:

• from minus infinity to 1;
• from 4 to 9.

This is fairly easy to determine. Substitute any number from the interval into the function and see what sign the answer is (minus or plus).

Function Ascending and Decreasing

In order to explore and build a function, we need to find out where the graph will increase (go up on Oy), and where it will fall (creep down along the y-axis).

The function increases only if the larger value of the variable x corresponds to greater value y. That is, x2 is greater than x1, and f(x2) is greater than f(x1). And we observe a completely opposite phenomenon in a decreasing function (the more x, the less y). To determine the intervals of increase and decrease, you need to find the following:

• scope (we already have it);
• derivative (in our case: 1/3(3x^2-28x+49);
• solve the equation 1/3(3x^2-28x+49)=0.

After calculations, we get the result:

We get: the function increases on the intervals from minus infinity to 7/3 and from 7 to infinity, and decreases on the interval from 7/3 to 7.

Extremes

The investigated function y=1/3(x^3-14x^2+49x-36) is continuous and exists for any values ​​of the variable x. The extremum point shows the maximum and minimum of this function. In our case, there are none, which greatly simplifies the construction task. Otherwise, they are also found using the derivative function. After finding, do not forget to mark them on the chart.

Convexity and concavity

We continue to study the function y(x). Now we need to check it for convexity and concavity. The definitions of these concepts are quite difficult to perceive, it is better to analyze everything with examples. For the test: a function is convex if it is a non-decreasing function. Agree, this is incomprehensible!

We need to find the derivative of the second order function. We get: y=1/3(6x-28). Now equate right side to zero and solve the equation. Answer: x=14/3. We have found the inflection point, that is, the place where the graph changes from convex to concave or vice versa. On the interval from minus infinity to 14/3, the function is convex, and from 14/3 to plus infinity, it is concave. It is also very important to note that the inflection point on the chart should be smooth and soft, no sharp corners should not be present.

Definition of additional points

Our task is to explore and plot the function graph. We have completed the study, it will not be difficult to plot the function now. For a more accurate and detailed reproduction of a curve or a straight line on the coordinate plane, you can find several auxiliary points. It's pretty easy to calculate them. For example, we take x=3, solve the resulting equation and find y=4. Or x=5 and y=-5 and so on. You can take as many additional points as you need to build. At least 3-5 of them are found.

Plotting

We needed to investigate the function (x^3-14x^2+49x-36)*1/3=y. All the necessary marks in the course of calculations were made on the coordinate plane. All that remains to be done is to build a graph, that is, connect all the points to each other. Connecting the dots is smooth and accurate, this is a matter of skill - a little practice and your schedule will be perfect.

How to investigate a function and plot its graph?

It seems that I am beginning to understand the soulful face of the leader of the world proletariat, the author of collected works in 55 volumes .... The long journey began with elementary information about functions and graphs, and now work on a laborious topic ends with a natural result - an article about the full function study. The long-awaited task is formulated as follows:

Investigate the function by methods of differential calculus and, based on the results of the study, build its graph

Or in short: examine the function and plot it.

Why explore? AT simple cases it will not be difficult for us to deal with elementary functions, draw a graph obtained using elementary geometric transformations etc. However, the properties and graphic representations of more complex functions are far from obvious, which is why a whole study is needed.

The main steps of the solution are summarized in the reference material Function Study Scheme, this is your section guide. Dummies need a step-by-step explanation of the topic, some readers don't know where to start and how to organize the study, and advanced students may be interested in only a few points. But whoever you are, dear visitor, the proposed summary with pointers to various lessons in the shortest time will orient and guide you in the direction of interest. The robots shed a tear =) The manual was made up in the form of a pdf file and took its rightful place on the page Mathematical formulas and tables.

I used to break the study of the function into 5-6 points:

6) Additional points and graph based on the results of the study.

As for the final action, I think everyone understands everything - it will be very disappointing if in a matter of seconds it is crossed out and the task is returned for revision. A CORRECT AND ACCURATE DRAWING is the main result of the solution! It is very likely to "cover up" analytical oversights, while an incorrect and/or sloppy schedule will cause problems even with a perfectly conducted study.

It should be noted that in other sources, the number of research items, the order of their implementation and the design style may differ significantly from the scheme proposed by me, but in most cases it is quite enough. The simplest version of the problem consists of only 2-3 stages and is formulated something like this: “explore the function using the derivative and plot” or “explore the function using the 1st and 2nd derivative, plot”.

Naturally, if another algorithm is analyzed in detail in your training manual or your teacher strictly requires you to adhere to his lectures, then you will have to make some adjustments to the solution. No more difficult than replacing a fork with a chainsaw spoon.

Let's check the function for even / odd:

This is followed by a template unsubscribe:
, so this function is neither even nor odd.

Since the function is continuous on , there are no vertical asymptotes.

There are no oblique asymptotes either.

Note : I remind you that the higher order of growth than , so the final limit is exactly " plus infinity."

Let's find out how the function behaves at infinity:

In other words, if we go to the right, then the graph goes infinitely far up, if we go to the left, infinitely far down. Yes, there are also two limits under a single entry. If you have difficulty deciphering the signs, please visit the lesson about infinitesimal functions.

So the function not limited from above and not limited from below. Considering that we do not have break points, it becomes clear and function range: is also any real number.

USEFUL TECHNIQUE

Each task step brings new information about the graph of the function, so in the course of the solution it is convenient to use a kind of LAYOUT. Let's draw a Cartesian coordinate system on the draft. What is known for sure? Firstly, the graph has no asymptotes, therefore, there is no need to draw straight lines. Second, we know how the function behaves at infinity. According to the analysis, we draw the first approximation:

Note that in effect continuity function on and the fact that , the graph must cross the axis at least once. Or maybe there are several points of intersection?

3) Zeros of the function and intervals of constant sign.

First, find the intersection point of the graph with the y-axis. It's simple. It is necessary to calculate the value of the function when:

Half above sea level.

To find the points of intersection with the axis (zeroes of the function), you need to solve the equation, and here an unpleasant surprise awaits us:

At the end, a free member lurks, which significantly complicates the task.

Such an equation has at least one real root, and most often this root is irrational. In the worst fairy tale, three little pigs are waiting for us. The equation is solvable using the so-called Cardano's formulas, but paper damage is comparable to almost the entire study. In this regard, it is wiser orally or on a draft to try to pick up at least one whole root. Let's check if these numbers are:
- does not fit;
- there is!

It's lucky here. In case of failure, you can also test and, and if these numbers do not fit, then I'm afraid there are very few chances for a profitable solution to the equation. Then it is better to skip the research point completely - maybe something will become clearer at the final step, when additional points will break through. And if the root (roots) are clearly “bad”, then it is better to remain modestly silent about the intervals of constancy of signs and to more accurately complete the drawing.

However, we have a beautiful root, so we divide the polynomial for no remainder:

The algorithm for dividing a polynomial by a polynomial is discussed in detail in the first example of the lesson. Complex Limits.

As a result, the left side of the original equation expands into a product:

And now a little about healthy way life. Of course I understand that quadratic equations need to be solved every day, but today we will make an exception: the equation has two real roots.

On the number line, we plot the found values and interval method define the signs of the function:


Thus, on the intervals chart located
below the x-axis, and at intervals - above this axis.

The resulting findings allow us to refine our layout, and the second approximation of the graph looks like this:

Please note that the function must have at least one maximum on the interval, and at least one minimum on the interval. But we don't know how many times, where and when the schedule will "wind around". By the way, a function can have infinitely many extremes.

4) Increasing, decreasing and extrema of the function.

Let's find the critical points:

This equation has two real roots. Let's put them on the number line and determine the signs of the derivative:


Therefore, the function increases by and decreases by .
At the point the function reaches its maximum: .
At the point the function reaches its minimum: .

The established facts drive our template into a rather rigid framework:

Needless to say, differential calculus is a powerful thing. Let's finally deal with the shape of the graph:

5) Convexity, concavity and inflection points.

Find the critical points of the second derivative:

Let's define signs:


The function graph is convex on and concave on . Let's calculate the ordinate of the inflection point: .

Almost everything cleared up.

6) It remains to find additional points that will help to more accurately build a graph and perform a self-test. In this case, they are few, but we will not neglect:

Let's execute the drawing:

in green the inflection point is marked, the crosses indicate additional points. Schedule cubic function is symmetrical about its inflection point, which is always located exactly in the middle between the maximum and minimum.

In the course of the assignment, I gave three hypothetical intermediate drawings. In practice, it is enough to draw a coordinate system, mark the points found, and after each point of the study, mentally figure out what the graph of the function might look like. Students with good level preparation, it will not be difficult to conduct such an analysis solely in the mind without involving a draft.

For a standalone solution:

Example 2

Explore the function and build a graph.

It's faster and more fun here. exemplary sample finishing touches at the end of the lesson.

A lot of secrets are revealed by the study of fractional rational functions:

Example 3

Using the methods of differential calculus, investigate the function and, based on the results of the study, construct its graph.

Decision: the first stage of the study does not differ in anything remarkable, with the exception of a hole in the definition area:

1) The function is defined and continuous on the entire number line except for the point , domain: .

, so this function is neither even nor odd.

Obviously, the function is non-periodic.

The graph of the function consists of two continuous branches located in the left and right half-plane - this is perhaps the most important conclusion of the 1st paragraph.

2) Asymptotes, the behavior of a function at infinity.

a) With the help of one-sided limits, we study the behavior of the function near the suspicious point, where the vertical asymptote must clearly be:

Indeed, the functions endure endless gap at the point
and the straight line (axis) is vertical asymptote graphic arts .

b) Check if oblique asymptotes exist:

Yes, the line is oblique asymptote graphics if .

It makes no sense to analyze the limits, since it is already clear that the function in an embrace with its oblique asymptote not limited from above and not limited from below.

The second point of the study brought a lot important information about the function. Let's do a rough sketch:

Conclusion No. 1 concerns intervals of sign constancy. At "minus infinity" the graph of the function is uniquely located below the x-axis, and at "plus infinity" it is above this axis. In addition, one-sided limits told us that both to the left and to the right of the point, the function is also greater than zero. Please note that in the left half-plane, the graph must cross the x-axis at least once. In the right half-plane, there may be no zeros of the function.

Conclusion No. 2 is that the function increases on and to the left of the point (goes “from bottom to top”). To the right of this point, the function decreases (goes “from top to bottom”). The right branch of the graph must certainly have at least one minimum. On the left, extremes are not guaranteed.

Conclusion No. 3 gives reliable information about the concavity of the graph in the vicinity of the point. We cannot yet say anything about convexity/concavity at infinity, since the line can be pressed against its asymptote both from above and from below. Generally speaking, there is an analytical way to figure this out right now, but the shape of the chart "for nothing" will become clearer at a later stage.

Why so many words? To control subsequent research points and avoid mistakes! Further calculations should not contradict the conclusions drawn.

3) Points of intersection of the graph with the coordinate axes, intervals of constant sign of the function.

The graph of the function does not cross the axis.

Using the interval method, we determine the signs:

, if ;
, if .

The results of the paragraph are fully consistent with Conclusion No. 1. After each step, look at the draft, mentally refer to the study, and finish drawing the graph of the function.

In this example, the numerator is divided term by term by the denominator, which is very beneficial for differentiation:

Actually, this has already been done when finding asymptotes.

- critical point.

Let's define signs:

increases by and decreases to

At the point the function reaches its minimum: .

There were also no discrepancies with Conclusion No. 2, and, most likely, we are on the right track.

This means that the graph of the function is concave over the entire domain of definition.

Excellent - and you don't need to draw anything.

There are no inflection points.

The concavity is consistent with Conclusion No. 3, moreover, it indicates that at infinity (both there and there) the graph of the function is located higher its oblique asymptote.

6) We will conscientiously pin the task with additional points. Here we have to work hard, because we know only two points from the study.

And a picture that, probably, many have long presented:

In the course of the assignment, care must be taken to ensure that there are no contradictions between the stages of the study, but sometimes the situation is urgent or even desperately dead-end. Here the analytics "does not converge" - and that's it. In this case, I recommend an emergency technique: we find as many points belonging to the graph as possible (how much patience is enough), and mark them on the coordinate plane. Graphical analysis of the found values ​​in most cases will tell you where is the truth and where is the lie. In addition, the graph can be pre-built using some program, for example, in the same Excel (it is clear that this requires skills).

Example 4

Using the methods of differential calculus, investigate the function and plot its graph.

This is a do-it-yourself example. In it, self-control is enhanced by the evenness of the function - the graph is symmetrical about the axis, and if something in your study contradicts this fact, look for an error.

Even or odd function can be investigated only for , and then use the symmetry of the graph. This solution is optimal, but it looks, in my opinion, very unusual. Personally, I consider the entire numerical axis, but I still find additional points only on the right:

Example 5

Conduct a complete study of the function and plot its graph.

Decision: rushed hard:

1) The function is defined and continuous on the entire real line: .

This means that this function is odd, its graph is symmetrical with respect to the origin.

Obviously, the function is non-periodic.

2) Asymptotes, the behavior of a function at infinity.

Since the function is continuous on , there are no vertical asymptotes

For a function containing an exponent, typically separate the study of "plus" and "minus infinity", however, our life is facilitated just by the symmetry of the graph - either there is an asymptote on the left and on the right, or it is not. Therefore, both infinite limits can be arranged under a single entry. In the course of the solution, we use L'Hopital's rule:

The straight line (axis) is the horizontal asymptote of the graph at .

Pay attention to how I cleverly avoided the full algorithm for finding the oblique asymptote: the limit is quite legal and clarifies the behavior of the function at infinity, and the horizontal asymptote was found "as if at the same time."

It follows from the continuity on and the existence of a horizontal asymptote that the function limited from above and limited from below.

3) Points of intersection of the graph with the coordinate axes, intervals of constancy.

Here we also shorten the solution:
The graph passes through the origin.

There are no other points of intersection with the coordinate axes. Moreover, the intervals of constancy are obvious, and the axis can not be drawn: , which means that the sign of the function depends only on the “x”:
, if ;
, if .

4) Increasing, decreasing, extrema of the function.


are critical points.

The points are symmetrical about zero, as it should be.

Let's define the signs of the derivative:


The function increases on the interval and decreases on the intervals

At the point the function reaches its maximum: .

Due to the property (oddity of the function) the minimum can be omitted:

Since the function decreases on the interval , then, obviously, the graph is located at "minus infinity" under with its asymptote. On the interval, the function also decreases, but here the opposite is true - after passing through the maximum point, the line approaches the axis from above.

It also follows from the above that the function graph is convex at "minus infinity" and concave at "plus infinity".

After this point of the study, the area of ​​\u200b\u200bvalues ​​of the function was also drawn:

If you have a misunderstanding of any points, I once again urge you to draw coordinate axes in your notebook and, with a pencil in your hands, re-analyze each conclusion of the task.

5) Convexity, concavity, inflections of the graph.

are critical points.

The symmetry of the points is preserved, and, most likely, we are not mistaken.

Let's define signs:


The graph of the function is convex on and concave on .

Convexity/concavity at extreme intervals was confirmed.

At all critical points there are inflections in the graph. Let's find the ordinates of the inflection points, while again reducing the number of calculations, using the oddness of the function:

The study of the function is carried out according to a clear scheme and requires the student to have solid knowledge of basic mathematical concepts such as the domain of definition and values, the continuity of the function, the asymptote, extremum points, parity, periodicity, etc. The student must freely differentiate functions and solve equations, which are sometimes very intricate.

That is, this task checks a significant layer of knowledge, any gap in which will become an obstacle to obtaining right decision. Especially often difficulties arise with the construction of graphs of functions. This mistake immediately catches the eye of the teacher and can greatly ruin your grade, even if everything else was done correctly. Here you can find tasks for the study of the function online: study examples, download solutions, order assignments.

Investigate a Function and Plot: Examples and Solutions Online

We have prepared for you a lot of ready-made feature studies, both paid in the solution book, and free in the Feature Research Examples section. On the basis of these solved tasks, you will be able to get acquainted in detail with the methodology for performing such tasks, by analogy, perform your own research.

We offer ready-made examples complete research and plotting of the function graph of the most common types: polynomials, fractional-rational, irrational, exponential, logarithmic, trigonometric functions. Each solved problem is accompanied by a ready-made graph with selected key points, asymptotes, maxima and minima, the solution is carried out according to the algorithm for studying the function.

Solved examples, in any case, will become for you good help, as they cover the most popular types of functions. We offer you hundreds of already solved problems, but, as you know, there are an infinite number of mathematical functions in the world, and teachers are great experts in inventing more and more intricate tasks for poor students. So, dear students, qualified assistance will not hurt you.

Solving problems for the study of a function to order

In this case, our partners will offer you another service - full function study online to order. The task will be completed for you in compliance with all the requirements for the algorithm for solving such problems, which will greatly please your teacher.

We will do a complete study of the function for you: we will find the domain of definition and the range of values, examine for continuity and discontinuity, set the parity, check your function for periodicity, find the points of intersection with the coordinate axes. And, of course, further with the help of differential calculus: we will find asymptotes, calculate extrema, inflection points, build the graph itself.

One of the most important tasks of differential calculus is the development common examples studies of the behavior of functions.

If the function y \u003d f (x) is continuous on the interval, and its derivative is positive or equal to 0 on the interval (a, b), then y \u003d f (x) increases by (f "(x) 0). If the function y \u003d f (x) is continuous on the segment , and its derivative is negative or equal to 0 on the interval (a,b), then y=f(x) decreases by (f"(x)0)

The intervals in which the function does not decrease or increase are called intervals of monotonicity of the function. The nature of the monotonicity of a function can change only at those points of its domain of definition, at which the sign of the first derivative changes. The points at which the first derivative of a function vanishes or breaks are called critical points.

Theorem 1 (1st sufficient condition the existence of an extremum).

Let the function y=f(x) be defined at the point x 0 and let there be a neighborhood δ>0 such that the function is continuous on the segment , differentiable on the interval (x 0 -δ, x 0)u(x 0 , x 0 + δ) , and its derivative retains a constant sign on each of these intervals. Then if on x 0 -δ, x 0) and (x 0, x 0 + δ) the signs of the derivative are different, then x 0 is an extremum point, and if they match, then x 0 is not an extremum point. Moreover, if, when passing through the point x0, the derivative changes sign from plus to minus (to the left of x 0, f "(x)> 0 is performed, then x 0 is the maximum point; if the derivative changes sign from minus to plus (to the right of x 0 is executed by f"(x)<0, то х 0 - точка минимума.

The maximum and minimum points are called the extremum points of the function, and the maxima and minima of the function are called its extreme values.

Theorem 2 (necessary criterion for a local extremum).

If the function y=f(x) has an extremum at the current x=x 0, then either f'(x 0)=0 or f'(x 0) does not exist.
At the extremum points of a differentiable function, the tangent to its graph is parallel to the Ox axis.

Algorithm for studying a function for an extremum:

1) Find the derivative of the function.
2) Find critical points, i.e. points where the function is continuous and the derivative is zero or does not exist.
3) Consider the neighborhood of each of the points, and examine the sign of the derivative to the left and right of this point.
4) Determine the coordinates of the extreme points, for this value of the critical points, substitute into this function. Using sufficient extremum conditions, draw appropriate conclusions.

Example 18. Investigate the function y=x 3 -9x 2 +24x

Decision.
1) y"=3x 2 -18x+24=3(x-2)(x-4).
2) Equating the derivative to zero, we find x 1 =2, x 2 =4. In this case, the derivative is defined everywhere; hence, apart from the two found points, there are no other critical points.
3) The sign of the derivative y "=3(x-2)(x-4) changes depending on the interval as shown in Figure 1. When passing through the point x=2, the derivative changes sign from plus to minus, and when passing through the point x=4 - from minus to plus.
4) At the point x=2, the function has a maximum y max =20, and at the point x=4 - a minimum y min =16.

Theorem 3. (2nd sufficient condition for the existence of an extremum).

Let f "(x 0) and f "" (x 0) exist at the point x 0. Then if f "" (x 0)> 0, then x 0 is the minimum point, and if f "" (x 0)<0, то х 0 – точка максимума функции y=f(x).

On the segment, the function y \u003d f (x) can reach the smallest (at least) or largest (at most) value either at the critical points of the function lying in the interval (a; b), or at the ends of the segment.

The algorithm for finding the largest and smallest values ​​of a continuous function y=f(x) on the segment :

1) Find f "(x).
2) Find the points at which f "(x) = 0 or f" (x) - does not exist, and select from them those that lie inside the segment.
3) Calculate the value of the function y \u003d f (x) at the points obtained in paragraph 2), as well as at the ends of the segment and choose the largest and smallest of them: they are, respectively, the largest (for the largest) and the smallest (for the smallest) function values ​​on the segment .

Example 19. Find the largest value of a continuous function y=x 3 -3x 2 -45+225 on the segment .

1) We have y "=3x 2 -6x-45 on the segment
2) The derivative y" exists for all x. Let's find the points where y"=0; we get:
3x2 -6x-45=0
x 2 -2x-15=0
x 1 \u003d -3; x2=5
3) Calculate the value of the function at the points x=0 y=225, x=5 y=50, x=6 y=63
Only the point x=5 belongs to the segment. The largest of the found values ​​of the function is 225, and the smallest is the number 50. So, at max = 225, at max = 50.

Investigation of a function on convexity

The figure shows the graphs of two functions. The first of them is turned with a bulge up, the second - with a bulge down.

The function y=f(x) is continuous on a segment and differentiable in the interval (a;b), is called convex up (down) on this segment if, for axb, its graph lies no higher (not lower) than the tangent drawn at any point M 0 (x 0 ;f(x 0)), where axb.

Theorem 4. Let the function y=f(x) have a second derivative at any interior point x of the segment and be continuous at the ends of this segment. Then if the inequality f""(x)0 is satisfied on the interval (a;b), then the function is downward convex on the segment ; if the inequality f""(x)0 is satisfied on the interval (а;b), then the function is convex upward on .

Theorem 5. If the function y \u003d f (x) has a second derivative on the interval (a; b) and if it changes sign when passing through the point x 0, then M (x 0 ; f (x 0)) is an inflection point.

Rule for finding inflection points:

1) Find points where f""(x) does not exist or vanishes.
2) Examine the sign f""(x) to the left and right of each point found at the first step.
3) Based on Theorem 4, draw a conclusion.

Example 20. Find extremum points and inflection points of the function graph y=3x 4 -8x 3 +6x 2 +12.

We have f"(x)=12x 3 -24x 2 +12x=12x(x-1) 2. Obviously, f"(x)=0 for x 1 =0, x 2 =1. The derivative, when passing through the point x=0, changes sign from minus to plus, and when passing through the point x=1, it does not change sign. This means that x=0 is the minimum point (y min =12), and there is no extremum at the point x=1. Next, we find . The second derivative vanishes at the points x 1 =1, x 2 =1/3. The signs of the second derivative change as follows: On the ray (-∞;) we have f""(x)>0, on the interval (;1) we have f""(x)<0, на луче (1;+∞) имеем f""(x)>0. Therefore, x= is the inflection point of the function graph (transition from convexity down to convexity up) and x=1 is also an inflection point (transition from convexity up to convexity down). If x=, then y= ; if, then x=1, y=13.

An algorithm for finding the asymptote of a graph

I. If y=f(x) as x → a , then x=a is a vertical asymptote.
II. If y=f(x) as x → ∞ or x → -∞ then y=A is the horizontal asymptote.
III. To find the oblique asymptote, we use the following algorithm:
1) Calculate . If the limit exists and is equal to b, then y=b is the horizontal asymptote; if , then go to the second step.
2) Calculate . If this limit does not exist, then there is no asymptote; if it exists and is equal to k, then go to the third step.
3) Calculate . If this limit does not exist, then there is no asymptote; if it exists and is equal to b, then go to the fourth step.
4) Write down the equation of the oblique asymptote y=kx+b.

Example 21: Find an asymptote for a function

1)
2)
3)
4) The oblique asymptote equation has the form

The scheme of the study of the function and the construction of its graph

I. Find the domain of the function.
II. Find the points of intersection of the graph of the function with the coordinate axes.
III. Find asymptotes.
IV. Find points of possible extremum.
V. Find critical points.
VI. Using the auxiliary drawing, investigate the sign of the first and second derivatives. Determine the areas of increase and decrease of the function, find the direction of the convexity of the graph, extremum points and inflection points.
VII. Build a graph, taking into account the study conducted in paragraphs 1-6.

Example 22: Plot a function graph according to the above scheme

Decision.
I. The domain of the function is the set of all real numbers, except for x=1.
II. Since the equation x 2 +1=0 has no real roots, then the graph of the function does not have points of intersection with the Ox axis, but intersects the Oy axis at the point (0; -1).
III. Let us clarify the question of the existence of asymptotes. We investigate the behavior of the function near the discontinuity point x=1. Since y → ∞ for x → -∞, y → +∞ for x → 1+, then the line x=1 is a vertical asymptote of the graph of the function.
If x → +∞(x → -∞), then y → +∞(y → -∞); therefore, the graph does not have a horizontal asymptote. Further, from the existence of limits

Solving the equation x 2 -2x-1=0, we get two points of a possible extremum:
x 1 =1-√2 and x 2 =1+√2

V. To find the critical points, we calculate the second derivative:

Since f""(x) does not vanish, there are no critical points.
VI. We investigate the sign of the first and second derivatives. Possible extremum points to be considered: x 1 =1-√2 and x 2 =1+√2, divide the area of ​​existence of the function into intervals (-∞;1-√2),(1-√2;1+√2) and (1+√2;+∞).

In each of these intervals, the derivative retains its sign: in the first - plus, in the second - minus, in the third - plus. The sequence of signs of the first derivative will be written as follows: +, -, +.
We get that the function on (-∞;1-√2) increases, on (1-√2;1+√2) it decreases, and on (1+√2;+∞) it increases again. Extremum points: maximum at x=1-√2, moreover f(1-√2)=2-2√2 minimum at x=1+√2, moreover f(1+√2)=2+2√2. On (-∞;1) the graph is convex upwards, and on (1;+∞) - downwards.
VII Let's make a table of the obtained values

VIII Based on the data obtained, we build a sketch of the graph of the function

The reference points in the study of functions and the construction of their graphs are characteristic points - points of discontinuity, extremum, inflection, intersection with the coordinate axes. With the help of differential calculus, it is possible to establish the characteristic features of the change in functions: increase and decrease, maxima and minima, the direction of the convexity and concavity of the graph, the presence of asymptotes.

A sketch of the function graph can (and should) be sketched after finding the asymptotes and extremum points, and it is convenient to fill in the summary table of the study of the function in the course of the study.

Usually, the following scheme of function research is used.

1.Find the domain, continuity intervals, and breakpoints of a function.

2.Examine the function for even or odd (axial or central symmetry of the graph.

3.Find asymptotes (vertical, horizontal or oblique).

4.Find and investigate the intervals of increase and decrease of the function, its extremum points.

5.Find the intervals of convexity and concavity of the curve, its inflection points.

6.Find the points of intersection of the curve with the coordinate axes, if they exist.

7.Compile a summary table of the study.

8.Build a graph, taking into account the study of the function, carried out according to the above points.

Example. Explore Function

and plot it.

7. Let's make a summary table of the study of the function, where we will enter all the characteristic points and the intervals between them. Given the parity of the function, we get the following table: