Intervals ascending and descending function online. Function research

Extreme functions

Definition 2

The point $ x_0 $ is called the maximum point of the function $ f (x) $ if there exists a neighborhood of this point such that for all $ x $ from this neighborhood the inequality $ f (x) \\ le f (x_0) $ holds.

Definition 3

The point $ x_0 $ is called the maximum point of the function $ f (x) $ if there is a neighborhood of this point such that for all $ x $ from this neighborhood the inequality $ f (x) \\ ge f (x_0) $ holds.

The concept of an extremum of a function is closely related to the concept of a critical point of a function. We introduce its definition.

Definition 4

$ x_0 $ is called the critical point of the function $ f (x) $ if:

1) $ x_0 $ - the internal point of the definition domain;

2) $ f "\\ left (x_0 \\ right) \u003d 0 $ or does not exist.

For the concept of an extremum, one can formulate theorems on sufficient and necessary conditions for its existence.

Theorem 2

A sufficient condition for extremum

Let the point $ x_0 $ be critical for the function $ y \u003d f (x) $ and lie in the interval $ (a, b) $. Let on each interval $ \\ left (a, x_0 \\ right) \\ and \\ (x_0, b) $ the derivative $ f "(x) $ exist and keep a constant sign. Then:

1) If on the interval $ (a, x_0) $ the derivative is $ f "\\ left (x \\ right)\u003e 0 $, and on the interval $ (x_0, b) $ the derivative is $ f" \\ left (x \\ right)

2) If on the interval $ (a, x_0) $ the derivative is $ f "\\ left (x \\ right) 0 $, then the point $ x_0 $ is the minimum point for this function.

3) If on the interval $ (a, x_0) $ and on the interval $ (x_0, b) $ the derivative $ f "\\ left (x \\ right)\u003e 0 $ or the derivative $ f" \\ left (x \\ right)

This theorem is illustrated in Figure 1.

Figure 1. A sufficient condition for the existence of extrema

Examples of extremes (Fig. 2).

Figure 2. Examples of extreme points

Rule of function research on extremum

2) Find the derivative $ f "(x) $;

7) Draw conclusions about the presence of maxima and minima in each interval using Theorem 2.

Function increase and decrease

We introduce, for starters, the definition of increasing and decreasing functions.

Definition 5

The function $ y \u003d f (x) $ defined on the interval $ X $ is called increasing if for any points $ x_1, x_2 \\ in X $ for $ x_1

Definition 6

The function $ y \u003d f (x) $ defined on the interval $ X $ is called decreasing if for any points $ x_1, x_2 \\ in X $ for $ x_1f (x_2) $.

Investigation of the function of increasing and decreasing

You can explore the functions of increasing and decreasing using the derivative.

In order to investigate the function for intervals of increase and decrease, it is necessary to do the following:

1) Find the domain of the function $ f (x) $;

2) Find the derivative $ f "(x) $;

3) Find the points at which the equality $ f "\\ left (x \\ right) \u003d 0 $ holds;

4) Find the points at which $ f "(x) $ does not exist;

5) Mark on the coordinate line all the points found and the domain of this function;

6) Determine the sign of the derivative $ f "(x) $ on each resulting interval;

7) To conclude: at intervals where $ f "\\ left (x \\ right) 0 $ the function increases.

Examples of tasks for studying functions of increasing, decreasing, and the presence of extreme points

Example 1

Investigate the function of increasing and decreasing, and the presence of maximum and minimum points: $ f (x) \u003d (2x) ^ 3-15x ^ 2 + 36x + 1 $

Since the first 6 points are the same, let’s start with them.

1) Scope - all real numbers;

2) $ f "\\ left (x \\ right) \u003d 6x ^ 2-30x + 36 $;

3) $ f "\\ left (x \\ right) \u003d 0 $;

\ \ \

4) $ f "(x) $ exists at all points in the domain of definition;

5) Coordinate line:

Figure 3

6) Determine the sign of the derivative $ f "(x) $ on each interval:

\\ \\ if for any pair of points x and x ", and ≤ x the inequality f(x) ≤ f (x "), and strictly increasing - if the inequality f (x) f(x ") Decreasing and strict decreasing of a function are defined in a similar way. For example, the function at = x 2 (fig. , a) strictly increases on the segment, and

(fig. , b) strictly decreases on this segment. Increasing functions are denoted by f (x), and decreasing f (x) ↓. In order to differentiable function f (x) was increasing on the segment [ a, b], it is necessary and sufficient that its derivative f"(x) was non-negative on [ a, b].

Along with the increase and decrease of the function on the segment, the increase and decrease of the function at the point are considered. Function at = f (x) is called increasing at the point x 0 if there is such an interval (α, β) containing the point x 0 that for any point x from (α, β), x\u003e x 0, the inequality f (x 0) ≤ f (x), and for any point x from (α, β), x 0, the inequality f (x) ≤ f (x 0). Similarly, a strict increase in the function at the point x 0. If f"(x 0) > 0, then the function f(x) strictly increases at the point x 0. If f (x) increases at each point of the interval ( a, b), then it increases in this interval.

S. B. Stechkin.

Great Soviet Encyclopedia. - M .: Soviet Encyclopedia. 1969-1978 .

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Extreme functions

Definition 2

Definition 3

The concept of an extremum of a function is closely related to the concept of a critical point of a function. We introduce its definition.

Definition 4

$ x_0 $ is called the critical point of the function $ f (x) $ if:

1) $ x_0 $ - the internal point of the definition domain;

2) $ f "\\ left (x_0 \\ right) \u003d 0 $ or does not exist.

For the concept of an extremum, one can formulate theorems on sufficient and necessary conditions for its existence.

Theorem 2

A sufficient condition for extremum

1) If on the interval $ (a, x_0) $ the derivative is $ f "\\ left (x \\ right)\u003e 0 $, and on the interval $ (x_0, b) $ the derivative is $ f" \\ left (x \\ right)

2) If on the interval $ (a, x_0) $ the derivative is $ f "\\ left (x \\ right) 0 $, then the point $ x_0 $ is the minimum point for this function.

3) If on the interval $ (a, x_0) $ and on the interval $ (x_0, b) $ the derivative $ f "\\ left (x \\ right)\u003e 0 $ or the derivative $ f" \\ left (x \\ right)

This theorem is illustrated in Figure 1.

Figure 1. A sufficient condition for the existence of extrema

Examples of extremes (Fig. 2).

Figure 2. Examples of extreme points

Rule of function research on extremum

2) Find the derivative $ f "(x) $;

7) Draw conclusions about the presence of maxima and minima in each interval using Theorem 2.

Function increase and decrease

We introduce, for starters, the definition of increasing and decreasing functions.

Definition 5

The function $ y \u003d f (x) $ defined on the interval $ X $ is called increasing if for any points $ x_1, x_2 \\ in X $ for $ x_1

Definition 6

The function $ y \u003d f (x) $ defined on the interval $ X $ is called decreasing if for any points $ x_1, x_2 \\ in X $ for $ x_1f (x_2) $.

Investigation of the function of increasing and decreasing

You can explore the functions of increasing and decreasing using the derivative.

In order to investigate the function for intervals of increase and decrease, it is necessary to do the following:

1) Find the domain of the function $ f (x) $;

2) Find the derivative $ f "(x) $;

3) Find the points at which the equality $ f "\\ left (x \\ right) \u003d 0 $ holds;

4) Find the points at which $ f "(x) $ does not exist;

5) Mark on the coordinate line all the points found and the domain of this function;

6) Determine the sign of the derivative $ f "(x) $ on each resulting interval;

7) To conclude: at intervals where $ f "\\ left (x \\ right) 0 $ the function increases.

Examples of tasks for studying functions of increasing, decreasing, and the presence of extreme points

Example 1

Investigate the function of increasing and decreasing, and the presence of maximum and minimum points: $ f (x) \u003d (2x) ^ 3-15x ^ 2 + 36x + 1 $

Since the first 6 points are the same, let’s start with them.

1) Scope - all real numbers;

2) $ f "\\ left (x \\ right) \u003d 6x ^ 2-30x + 36 $;

3) $ f "\\ left (x \\ right) \u003d 0 $;

\ \ \

4) $ f "(x) $ exists at all points in the domain of definition;

5) Coordinate line:

Figure 3

6) Determine the sign of the derivative $ f "(x) $ on each interval:

\ \.

The range of function values \u200b\u200bis interval [1; 3].

1. For x \u003d -3, x \u003d - 1, x \u003d 1.5, x \u003d 4.5, the value of the function is zero.

The value of the argument at which the value of the function is zero is called the zero of the function.

//those. for this function, the numbers -3; -1; 1.5; 4,5 are zeros.

2. At intervals [4,5; 3) and (1; 1,5) and (4,5; 5,5], the graph of the function f is located above the abscissa axis, and at intervals (-3; -1) and (1,5; 4,5) below the axis abscissa, this is explained by the fact that on the intervals [4,5; 3) and (1; 1,5) and (4,5; 5,5] the function takes positive values, and on the intervals (-3; -1) and ( 1,5; 4,5) negative.

Each of the indicated intervals (where the function takes values \u200b\u200bof the same sign) is called the constant sign interval of the function f.//t.e. for example, if we take the interval (0; 3), then it is not an interval of the constant sign of this function.

In mathematics, it is customary to indicate intervals of maximum length when searching for intervals of constant sign of a function. //Those. the gap (2; 3) is constant sign function f, but the answer should include the interval [4,5; 3) containing the gap (2; 3).

3. If you move along the abscissa from 4.5 to 2, you will notice that the graph of the function goes down, that is, the values \u200b\u200bof the function decrease. // In mathematics, it is customary to say that in the interval [4,5; 2] the function decreases.

As x increases from 2 to 0, the function graph goes up, i.e. function values \u200b\u200bincrease. // In mathematics, it is customary to say that in the interval [2; 0] function increases.

A function f is called if for any two values \u200b\u200bof the argument x1 and x2 from this interval such that x2\u003e x1, the inequality f (x2)\u003e f (x1) holds. // or Function called increasing at some intervalif, for any values \u200b\u200bof the argument from this interval, a larger value of the function corresponds to a larger value of the function. // i.e. the more x, the more y.

The function f is called diminishing at some intervalif, for any two values \u200b\u200bof the argument x1 and x2 from this interval such that x2\u003e x1, the inequality f (x2) decreases on some interval, if for any values \u200b\u200bof the argument from this interval the larger value of the argument corresponds to a smaller value of the function. //those. the more x, the less y.

If the function increases over the entire domain of definition, then it is called increasing.

If the function decreases over the entire domain, then it is called diminishing.

Example 1 the graph of increasing and decreasing functions, respectively.

Example 2

Define Is the linear function f (x) \u003d 3x + 5 increasing or decreasing?

Evidence. We use the definitions. Let x1 and x2 be arbitrary values \u200b\u200bof the argument, with x1< x2., например х1=1, х2=7

Based on sufficient signs, there are intervals of increasing and decreasing functions.

Here are the wording of the signs:

if the derivative of the function y \u003d f (x) positive for any x from interval X, then the function increases by X;
if the derivative of the function y \u003d f (x) negative for any x from interval X, then the function decreases on X.

Thus, to determine the intervals of increase and decrease of the function, it is necessary:

find the scope of the function;

find the derivative of the function;

to the obtained intervals add boundary points at which the function is defined and continuous.

Consider an example to explain the algorithm.

Example.

Find the intervals of increasing and decreasing functions.

Decision.

The first step is to find a growth definition of the function. In our example, the expression in the denominator should not vanish, therefore, .

We pass to the derivative function:

To determine the intervals of increasing and decreasing functions by a sufficient criterion, we solve the inequalities and on the field of definition. We use a generalization of the interval method. The only valid numerator root is x \u003d 2, and the denominator vanishes at x \u003d 0. These points divide the domain into intervals at which the derivative of the function retains its sign. We mark these points on the number line. The pluses and minuses will arbitrarily denote the intervals at which the derivative is positive or negative. The arrows below schematically show the increase or decrease of the function in the corresponding interval.

Thus, and .

At the point x \u003d 2 the function is defined and continuous; therefore, it should be added to both the increase and the decrease. At the point x \u003d 0 the function is not defined, therefore this point is not included in the required intervals.

We give a graph of the function for comparing the results obtained with it.

Answer: function increases with decreases on the interval (0; 2] .

- Extremum points of a function of one variable. Sufficient extremum conditions

Suppose that a function f (x), defined and continuous in an interval, is not monotonic in it. There will be such parts [,] of the interval in which the largest and smallest value is reached by the function at the internal point, ie between and.

It is said that a function f (x) has a maximum (or minimum) at a point if this point can be surrounded by such a neighborhood (x 0 -, x 0 +) contained in the interval where the function is given that the inequality holds for all its points.

f (x)< f(x 0)(или f(x)>f (x 0))

In other words, the point x 0 delivers the maximum (minimum) to the function f (x) if the value f (x 0) turns out to be the largest (smallest) of the values \u200b\u200baccepted by the function in some (at least small) neighborhood of this point. Note that the very definition of maximum (minimum) assumes that the function is given on both sides of the point x 0.

If there exists a neighborhood within which (for x \u003d x 0) the strict inequality

f (x) f (x 0)

they say that the function has a maximum (minimum) at x 0, otherwise it is improper.

If the function has maxima at the points x 0 and x 1, then, applying the second Weierstrass theorem to the interval, we see that the function reaches its least value in this interval at some point x 2 between x 0 and x 1 and has a minimum there. Similarly, between two minima there will certainly be a maximum. In that simplest (and in practice, the most important) case, when a function generally has only a finite number of maxima and minima, they simply alternate.

Note that to denote the maximum or minimum, there is a term that unites them - the extremum.

The concepts of maximum (max f (x)) and minimum (min f (x)) are local properties of the function and take place at a certain point x 0. The concepts of the largest (sup f (x)) and smallest (inf f (x)) values \u200b\u200brefer to a finite interval and are global properties of a function on the interval.

Figure 1 shows that at points x 1 and x 3 there are local maxima, and at points x 2 and x 4 there are local minima. However, the function reaches the smallest value at x \u003d a, and the largest at x \u003d b.

We pose the problem of finding all the values \u200b\u200bof an argument that deliver extremum functions. In solving it, the derivative will play its main role.

Assume first that for the function f (x) in the interval (a, b) there exists a finite derivative. If at a point x 0 the function has an extremum, then applying to the interval (x 0 -, x 0 +), which was discussed above, Fermat's theorem, we conclude that f (x) \u003d 0 this is the necessary condition for the extremum. The extremum should be sought only at those points where the derivative is equal to zero.

It should not be thought, however, that every point at which the derivative is equal to zero delivers extremum functions: the condition just indicated is not sufficient