Find the intervals of convexity of the function. Convexity and Concavity Intervals of a Function Plot

Using the online calculator, you can find inflection points and convexity intervals of the function graph with the design of the solution in Word. Whether a function of two variables f (x1, x2) is convex is solved using the Hesse matrix.

Function entry rules:

The direction of the convexity of the graph of the function. Inflection points

Definition: A curve y = f (x) is called convex downward in the interval (a; b) if it lies above the tangent at any point of this interval.

Definition: A curve y = f (x) is called convex upward in the interval (a; b) if it lies below the tangent at any point of this interval.

Definition: The intervals in which the graph of a function is turned upward or downward, are called the intervals of the convexity of the graph of the function.

The downward or upward convexity of the curve, which is the graph of the function y = f (x), is characterized by the sign of its second derivative: if in some interval f ’’ (x)> 0, then the curve is convex downward in this interval; if f ’’ (x)< 0, то кривая выпукла вверх на этом промежутке.

Definition: The point of the graph of the function y = f (x), separating the intervals of the convexity of the opposite directions of this graph, is called the inflection point.

Only critical points of the second kind can serve as inflection points, i.e. points belonging to the domain of definition of the function y = f (x), at which the second derivative f '' (x) vanishes or has a discontinuity.

The rule for finding the inflection points of the graph of the function y = f (x)

1. Find the second derivative f '' (x).
2. Find critical points of the second kind of the function y = f (x), i.e. the point at which f '' (x) vanishes or breaks.
3. Investigate the sign of the second derivative f '' (x) in the interval into which the found critical points divide the domain of definition of the function f (x). If in this case the critical point x 0 separates the intervals of the convexity of opposite directions, then x 0 is the abscissa of the inflection point of the graph of the function.
4. Calculate the values ​​of the function at the inflection points.

Example 1. Find the intervals of convexity and inflection points of the following curve: f (x) = 6x 2 –x 3.
Solution: Find f ’(x) = 12x - 3x 2, f’ ’(x) = 12 - 6x.
Find the critical points by the second derivative by solving the equation 12-6x = 0. x = 2.


f (2) = 6 * 2 2 - 2 3 = 16
Answer: The function is convex upward for x∈ (2; + ∞); the function is convex downward for x∈ (-∞; 2); inflection point (2; 16).

Example 2. Does the function have inflection points: f (x) = x 3 -6x 2 + 2x-1

Example 3. Find the intervals on which the graph of the function is convex and curved: f (x) = x 3 -6x 2 + 12x + 4

To determine the convexity (concavity) of a function on a certain interval, the following theorems can be used.

Theorem 1. Let the function be defined and continuous on the interval and have a finite derivative. For a function to be convex (concave) in, it is necessary and sufficient that its derivative decrease (increase) on this interval.

Theorem 2. Let the function be defined and continuous together with its derivative on and has a continuous second derivative inside. For the convexity (concavity) of the function in it is necessary and sufficient that inside

Let us prove Theorem 2 for the case of convexity of a function.

Need. Let's take an arbitrary point. Expand the function near a point in a Taylor series

Equation of the tangent to a curve at a point with an abscissa:

Then the excess of the curve over the tangent to it at the point is equal to

Thus, the remainder is equal to the excess of the curve over the tangent to it at a point. By virtue of continuity, if , then also for, belonging to a sufficiently small neighborhood of the point, and therefore, obviously, for any value other than the value belonging to the indicated neighborhood.

Hence, the graph of the function lies above the tangent line and the curve is convex at an arbitrary point.

Adequacy. Let the curve be convex on the interval. Let's take an arbitrary point.

Similarly to the previous one, we expand the function near a point into a Taylor series

The excess of the curve over the tangent to it at the point having the abscissa, determined by the expression is

Since the excess is positive for a sufficiently small neighborhood of the point, the second derivative is also positive. As we strive, we obtain that for an arbitrary point .

Example. Explore the convexity (concavity) function.

Its derivative increases on the entire number axis, which means that by Theorem 1 the function is concave on.

Its second derivative , therefore, by Theorem 2, the function is concave on.

3.4.2.2 Inflection points

Definition. Inflection point the graph of a continuous function is called the point that separates the intervals in which the function is convex and concave.

It follows from this definition that the inflection points are the extremum points of the first derivative. This implies the following statements for the necessary and sufficient inflection conditions.

Theorem (necessary inflection condition)... In order for a point to be an inflection point of a twice differentiable function, it is necessary that its second derivative at this point is equal to zero ( ) or did not exist.

Theorem (sufficient inflection condition). If the second derivative of a twice differentiable function changes sign when passing through a certain point, that is, an inflection point.

Note that the second derivative may not exist at the point itself.

The geometric interpretation of the inflection points is illustrated in Fig. 3.9

In a neighborhood of a point, the function is convex and its graph lies below the tangent drawn at this point. In the vicinity of a point, the function is concave and its graph lies above the tangent drawn at this point. At the inflection point, the tangent divides the graph of the function into areas of convexity and concavity.

3.4.2.3 Investigation of the function for convexity and the presence of inflection points

1. Find the second derivative.

2. Find the points at which the second derivative or does not exist.

Rice. 3.9.

3. Investigate the sign of the second derivative to the left and right of the found points and draw a conclusion about the intervals of convexity or concavity and the presence of inflection points.

Example. Examine the function for convexity and the presence of inflection points.

2. The second derivative is equal to zero at.

3. The second derivative changes sign at, so the point is the inflection point.

On an interval, then the function is convex on that interval.

On the interval, then the function is concave on this interval.

3.4.2.4 General scheme of the study of functions and plotting

When examining a function and plotting its graph, it is recommended to use the following scheme:

1. Find the domain of the function.
2. Investigate the function for evenness - oddness. Recall that the graph of an even function is symmetric about the ordinate axis, and the graph of an odd function is symmetric about the origin.
3. Find the vertical asymptotes.
4. Explore the behavior of a function at infinity, find horizontal or oblique asymptotes.
5. Find extrema and intervals of monotonicity of the function.
6. Find the convexity intervals of the function and the inflection points.
7. Find the points of intersection with the coordinate axes.

The study of the function is carried out simultaneously with the construction of its graph.

Example. Explore function and build her schedule.

1. Function definition area -.

2. The investigated function is even , therefore, its graph is symmetrical about the ordinate axis.

3. The denominator of the function vanishes at, so the graph of the function has vertical asymptotes and.

The points are discontinuity points of the second kind, since the limits on the left and right at these points tend to.

4. Behavior of the function at infinity.

Therefore, the graph of the function has a horizontal asymptote.

5. Extrema and intervals of monotony. Find the first derivative

For, therefore, the function decreases in these intervals.

For, therefore, the function increases in these intervals.

When, therefore, the point is the critical point.

Find the second derivative

Since, the point is the minimum point of the function.

6. Intervals of convexity and points of inflection.

Function at , so on this interval the function is concave.

The function at, means on these intervals the function is convex.

The function does not vanish anywhere, so there are no inflection points.

7. Points of intersection with coordinate axes.

The equation has a solution, which means the point of intersection of the function graph with the ordinate axis (0, 1).

The equation has no solution, so there are no points of intersection with the abscissa axis.

Taking into account the conducted research, it is possible to build a graph of the function

Schematic graph of a function shown in Fig. 3.10.

Rice. 3.10.

3.4.2.5 Asymptotes of the graph of a function

Definition. Asymptote the graph of a function is called a straight line, which has the property that the distance from a point () to this straight line tends to 0 with an unlimited distance from the origin of the graph point.

The general scheme of the study of the function and the construction of the graph.
1. Investigation of the function for convexity and concavity.

1. 
   
2. 
   Asymptotes of the graph of a function.

Introduction.

In your school mathematics course, you have already met with the need to plot graphs of functions. In, you used the point-by-point construction method. It should be noted that it is simple in concept and relatively quickly leads to the goal. In cases where the function is continuous and changes rather smoothly, this method can provide the necessary degree of accuracy of the graphical representation. To do this, you need to take more points in order to achieve a certain density of their placement.

Suppose now that the function in some places has features in its "behavior": either its values ​​somewhere in a small area change abruptly, or there are discontinuities. The most significant parts of the graph may not be detected in this way.

This circumstance also reduces the value of the method of plotting the graph "by points".

There is a second way to build charts, based on the analytical study of functions. It compares favorably with the method discussed in the school mathematics course.

1. Study of the function for convexity and concavity .

Let the function
differentiable on the interval (a, c). Then there is a tangent to the graph of the function at any point
this graph (
), and the tangent is not parallel to the OY axis, since its slope equal to
, of course.

O
assignment We will say that the graph of the function
on (a, b) has a downward (up) release, if it is located not below (not above) any tangent to the graph of the function on (a, b).

a) concave curve b) convex curve

Theorem 1 (a necessary condition for the convexity (concavity) of a curve).

If the graph of a twice differentiable function is a convex (concave) curve, then the second derivative on the interval (a, b) is negative (positive) on this interval.

Theorem 2(a sufficient condition for the convexity (concavity) of a curve).

If the function is twice differentiable on (a, b) and
(
) at all points of this interval, then the curve that is the graph of the function is convex (concave) on this interval.

1. 
   Inflection points of the function graph.

Definition Point
is called the inflection point of the graph of the function, if at the point
the graph has a tangent line, and there is a neighborhood of the point , within which the graph of the function on the left and right of the point has different directions of convexity.

O It can be seen that at the inflection point the tangent intersects the graph of the function, since on one side of this point the graph lies above the tangent, and on the other - below it, that is, in the vicinity of the inflection point, the graph of the function geometrically goes from one side of the tangent to the other and "Bends" over it. This is where the name "inflection point" comes from.

Theorem 3(a prerequisite for the inflection point). Let the graph of the function have an inflection at the point and let the function have at the point continuous second derivative. Then
.
Not every point for which is an inflection point. For example, the graph of the function
has no inflection at the point (0, 0), although
at
... Therefore, the equality to zero of the second derivative is only a necessary inflection condition.

The points of the graph for which it is called critical pointsII-cities. It is necessary to further investigate the issue of the presence of inflections at each critical point.

Theorem 4(a sufficient condition for an inflection point). Let the function have the second derivative in some neighborhood of the point. Then, if within the specified neighborhood
has different signs to the left and right of the point, then the graph has an inflection point at the point.
Comment. The theorem remains true if
has a second derivative in some neighborhood of the point, except for the point itself, and there is a tangent to the graph of the function at the point
... Then, if within the specified neighborhood has different signs to the left and right of the point, then the graph to the function has an inflection point at the point.
Scheme of the study of the function for convexity, concavity, inflection points.

Example. Explore function
on convexity, concavity, inflection points.
1.

2.
,
=

3. does not exist for

	)	1	(1, +)
	-		+
		1

1. 
   Asymptotes of the graph of a function.

When studying the behavior of a function with
or near discontinuity points of the 2nd kind, it often turns out that the graph of the function approaches as closely as possible to one or another straight line. Such straight lines are called.

O definition 1. Straight is called the asymptote of the curve L if the distance from the point of the curve to this straight line tends to zero as the point moves away along the curve to infinity. There are three types of asymptotes: vertical, horizontal, oblique.

Definition 2. Straight
is called the vertical asymptote of the graph of the function if at least one of the one-sided limits is
, that is, or

For example, the graph of the function
has a vertical asymptote
, because.
, a
.

Definition 3. The straight line y = A is called the horizontal asymptote of the graph of the function at
if
.

For example, the graph of a function has a horizontal asymptote y = 0, since.
.

Definition 4. Straight
(
) is called the oblique asymptote of the graph of the function at
if
;

If at least one of the limits does not exist, then the curve has no asymptotes. If, then these limits should be sought separately, for and
.

For example. Find the asymptotes of the graph of a function

; x = 0 - vertical asymptote

;
.

- oblique asymptote.
4. Scheme of a complete study of the function and plotting.

Let's consider an approximate scheme according to which it is advisable to investigate the behavior of a function and build its graph.

Example. Explore function
and build her schedule.

1., except for x = -1.

2.
function is neither even nor odd

-

-



+

+

y

-4


t p.

0

Conclusion.
An important feature of the considered method is that it is based primarily on the detection and study of characteristic features in the behavior of the curve. The places where the function changes smoothly are not studied in particular detail, and there is no need for such a study. But those places where the function has any peculiarities in behavior are subject to complete research and the most accurate graphic representation. These features are the points of maximum, minimum, points of discontinuity of the function, etc.

Determination of the direction of concavity and bends, as well as the indicated method of finding the asymptotes, make it possible to study functions in even more detail and get a more accurate idea of ​​their graphs.

Instructions

The inflection points of the function must belong to the domain of its definition, which must be found first. The graph of a function is a line that can be continuous or have discontinuities, decrease or increase monotonically, have minimum or maximum points (asymptotes), be convex or concave. An abrupt change in the last two states is called an inflection.

A necessary condition for the existence of an inflection of a function is the equality of the second to zero. Thus, by differentiating the function twice and equating the resulting expression to zero, one can find the abscissas of possible inflection points.

This condition follows from the definition of the properties of convexity and concavity of the graph of a function, i.e. negative and positive values ​​of the second derivative. At the inflection point, there is a sharp change in these properties, which means that the derivative goes over the zero mark. However, equality to zero is still not enough to denote an inflection.

There are two sufficient that the abscissa found at the previous stage belongs to the inflection point: Through this point, you can draw a tangent to the function. The second derivative has different signs to the right and left of the assumed inflection point. Thus, its existence at the point itself is not necessary, it is enough to determine that it changes sign at it. The second derivative of the function is equal to zero, and the third is not.

The first sufficient condition is universal and is used more often than others. Consider an illustrative example: y = (3 x + 3) ∛ (x - 5).

Solution: Find the scope. In this case, there are no restrictions, therefore, it is the entire space of real numbers. Calculate the first derivative: y '= 3 ∛ (x - 5) + (3 x + 3) / ∛ (x - 5) ².

Pay attention to the appearance of the fraction. It follows from this that the range of definition of the derivative is limited. The point x = 5 is punctured, which means that a tangent can pass through it, which partly corresponds to the first sign of the sufficiency of the inflection.

Determine the one-sided limits for the resulting expression as x → 5 - 0 and x → 5 + 0. They are -∞ and + ∞. You proved that a vertical tangent passes through the point x = 5. This point may turn out to be an inflection point, but first calculate the second derivative: Y '' = 1 / ∛ (x - 5) ² + 3 / ∛ (x - 5) ² - 2/3 (3 x + 3) / ∛ (x - 5) ^ 5 = (2 x - 22) / ∛ (x - 5) ^ 5.

Omit the denominator, since you have already taken into account the point x = 5. Solve the equation 2 x - 22 = 0. It has a single root x = 11. The last step is to confirm that the points x = 5 and x = 11 are inflection points. Analyze the behavior of the second derivative in their vicinity. It is obvious that at the point x = 5 it changes its sign from "+" to "-", and at the point x = 11 - vice versa. Conclusion: both points are inflection points. The first sufficient condition is satisfied.