The largest and smallest values ​​of a function of two variables in a closed region. Largest and smallest value of a function

From a practical point of view, the greatest interest is in using the derivative to find the largest and smallest values ​​of a function. What is this connected with? Maximizing profits, minimizing costs, determining the optimal load of equipment... In other words, in many areas of life we ​​have to solve problems of optimizing some parameters. And these are the tasks of finding the largest and smallest values ​​of a function.

It should be noted that the largest and smallest values ​​of a function are usually sought on a certain interval X, which is either the entire domain of the function or part of the domain of definition. The interval X itself can be a segment, an open interval , an infinite interval.

In this article we will talk about finding the largest and smallest values ​​of an explicitly defined function of one variable y=f(x) .

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The largest and smallest value of a function - definitions, illustrations.

Let's briefly look at the main definitions.

The largest value of the function that for anyone inequality is true.

The smallest value of the function y=f(x) on the interval X is called such a value that for anyone inequality is true.

These definitions are intuitive: the largest (smallest) value of a function is the largest (smallest) accepted value on the interval under consideration at the abscissa.

Stationary points– these are the values ​​of the argument at which the derivative of the function becomes zero.

Why do we need stationary points when finding the largest and smallest values? The answer to this question is given by Fermat's theorem. From this theorem it follows that if a differentiable function has an extremum (local minimum or local maximum) at some point, then this point is stationary. Thus, the function often takes its largest (smallest) value on the interval X at one of the stationary points from this interval.

Also, a function can often take on its largest and smallest values ​​at points at which the first derivative of this function does not exist, and the function itself is defined.

Let’s immediately answer one of the most common questions on this topic: “Is it always possible to determine the largest (smallest) value of a function”? No not always. Sometimes the boundaries of the interval X coincide with the boundaries of the domain of definition of the function, or the interval X is infinite. And some functions at infinity and at the boundaries of the domain of definition can take on both infinitely large and infinitely small values. In these cases, nothing can be said about the largest and smallest value of the function.

For clarity, we will give a graphic illustration. Look at the pictures and a lot will become clearer.

On the segment

In the first figure, the function takes the largest (max y) and smallest (min y) values ​​at stationary points located inside the segment [-6;6].

Consider the case depicted in the second figure. Let's change the segment to . In this example, the smallest value of the function is achieved at a stationary point, and the largest at the point with the abscissa corresponding to the right boundary of the interval.

In Figure 3, the boundary points of the segment [-3;2] are the abscissas of the points corresponding to the largest and smallest value of the function.

On an open interval

In the fourth figure, the function takes the largest (max y) and smallest (min y) values ​​at stationary points located inside the open interval (-6;6).

On the interval , no conclusions can be drawn about the largest value.

At infinity

In the example presented in the seventh figure, the function takes the largest value (max y) at a stationary point with abscissa x=1, and the smallest value (min y) is achieved on the right boundary of the interval. At minus infinity, the function values ​​asymptotically approach y=3.

Over the interval, the function reaches neither the smallest nor the largest value. As x=2 approaches from the right, the function values ​​tend to minus infinity (the line x=2 is a vertical asymptote), and as the abscissa tends to plus infinity, the function values ​​asymptotically approach y=3. A graphic illustration of this example is shown in Figure 8.

Algorithm for finding the largest and smallest values ​​of a continuous function on a segment.

Let us write an algorithm that allows us to find the largest and smallest values ​​of a function on a segment.

1. We find the domain of definition of the function and check whether it contains the entire segment.
2. We find all the points at which the first derivative does not exist and which are contained in the segment (usually such points are found in functions with an argument under the modulus sign and in power functions with a fractional-rational exponent). If there are no such points, then move on to the next point.
3. We determine all stationary points falling within the segment. To do this, we equate it to zero, solve the resulting equation and select suitable roots. If there are no stationary points or none of them fall into the segment, then move on to the next point.
4. We calculate the values ​​of the function at selected stationary points (if any), at points at which the first derivative does not exist (if any), as well as at x=a and x=b.
5. From the obtained values ​​of the function, we select the largest and smallest - they will be the required largest and smallest values ​​of the function, respectively.

Let's analyze the algorithm for solving an example to find the largest and smallest values ​​of a function on a segment.

Example.

Find the largest and smallest value of a function

• on the segment ;
• on the segment [-4;-1] .

Solution.

The domain of definition of a function is the entire set of real numbers, with the exception of zero, that is. Both segments fall within the definition domain.

Find the derivative of the function with respect to:

Obviously, the derivative of the function exists at all points of the segments and [-4;-1].

We determine stationary points from the equation. The only real root is x=2. This stationary point falls into the first segment.

For the first case, we calculate the values ​​of the function at the ends of the segment and at the stationary point, that is, for x=1, x=2 and x=4:

Therefore, the greatest value of the function is achieved at x=1, and the smallest value – at x=2.

For the second case, we calculate the function values ​​only at the ends of the segment [-4;-1] (since it does not contain a single stationary point):

Problem statement 2:

Given a function that is defined and continuous on a certain interval. You need to find the largest (smallest) value of the function on this interval.

Theoretical basis.
Theorem (Second Weierstrass Theorem):

If a function is defined and continuous in a closed interval, then it reaches its maximum and minimum values ​​in this interval.

The function can reach its largest and smallest values ​​either at the internal points of the interval or at its boundaries. Let's illustrate all the possible options.

Explanation:
1) The function reaches its greatest value on the left boundary of the interval at point , and its minimum value on the right boundary of the interval at point .
2) The function reaches its greatest value at the point (this is the maximum point), and its minimum value at the right boundary of the interval at the point.
3) The function reaches its maximum value on the left boundary of the interval at point , and its minimum value at point (this is the minimum point).
4) The function is constant on the interval, i.e. it reaches its minimum and maximum values ​​at any point in the interval, and the minimum and maximum values ​​are equal to each other.
5) The function reaches its greatest value at point , and its minimum value at point (despite the fact that the function has both a maximum and a minimum on this interval).
6) The function reaches its greatest value at a point (this is the maximum point), and its minimum value at a point (this is the minimum point).
Comment:

“Maximum” and “maximum value” are different things. This follows from the definition of maximum and the intuitive understanding of the phrase “maximum value”.

Algorithm for solving problem 2.



4) Select the largest (smallest) from the obtained values ​​and write down the answer.

Example 4:

Determine the largest and smallest value of a function on the segment.
Solution:
1) Find the derivative of the function.

2) Find stationary points (and points suspected of extremum) by solving the equation. Pay attention to the points at which there is no two-sided finite derivative.

3) Calculate the values ​​of the function at stationary points and at the boundaries of the interval.

4) Select the largest (smallest) from the obtained values ​​and write down the answer.

The function on this segment reaches its greatest value at the point with coordinates .

The function on this segment reaches its minimum value at the point with coordinates .

You can verify the correctness of the calculations by looking at the graph of the function under study.


Comment: The function reaches its greatest value at the maximum point, and its minimum at the boundary of the segment.

A special case.

Suppose you need to find the maximum and minimum values ​​of some function on a segment. After completing the first point of the algorithm, i.e. calculating the derivative, it becomes clear that, for example, it takes only negative values ​​throughout the entire interval under consideration. Remember that if the derivative is negative, then the function decreases. We found that the function decreases over the entire segment. This situation is shown in graph No. 1 at the beginning of the article.

The function decreases on the segment, i.e. it has no extrema points. From the picture you can see that the function will take the smallest value on the right boundary of the segment, and the largest value on the left. if the derivative on the segment is positive everywhere, then the function increases. The smallest value is on the left border of the segment, the largest is on the right.

The study of such an object of mathematical analysis as a function is of great importance meaning and in other fields of science. For example, in economic analysis there is a constant need to evaluate behavior functions profit, namely to determine its greatest meaning and develop a strategy to achieve it.

Instructions

The study of any behavior should always begin with a search for the domain of definition. Usually, according to the conditions of a specific problem, it is necessary to determine the largest meaning functions either over this entire area, or over a specific interval of it with open or closed borders.

Based on , the largest is meaning functions y(x0), in which for any point in the domain of definition the inequality y(x0) ≥ y(x) (x ≠ x0) holds. Graphically, this point will be the highest if the argument values ​​are placed along the abscissa axis, and the function itself along the ordinate axis.

To determine the greatest meaning functions, follow the three-step algorithm. Please note that you must be able to work with one-sided and , as well as calculate the derivative. So, let some function y(x) be given and you need to find its greatest meaning on a certain interval with boundary values ​​A and B.

Find out whether this interval is within the scope of the definition functions. To do this, you need to find it by considering all possible restrictions: the presence of a fraction, square root, etc. in the expression. The domain of definition is the set of argument values ​​for which the function makes sense. Determine whether the given interval is a subset of it. If yes, then move on to the next step.

Find the derivative functions and solve the resulting equation by equating the derivative to zero. This way you will get the values ​​of the so-called stationary points. Evaluate whether at least one of them belongs to the interval A, B.

At the third stage, consider these points and substitute their values ​​into the function. Depending on the interval type, perform the following additional steps. If there is a segment of the form [A, B], the boundary points are included in the interval; this is indicated by parentheses. Calculate Values functions for x = A and x = B. If the interval is open (A, B), the boundary values ​​are punctured, i.e. are not included in it. Solve one-sided limits for x→A and x→B. A combined interval of the form [A, B) or (A, B), one of whose boundaries belongs to it, the other does not. Find the one-sided limit as x tends to the punctured value, and substitute the other into the function. Infinite two-sided interval (-∞, +∞) or one-sided infinite intervals of the form: , (-∞, B).For real limits A and B, proceed according to the principles already described, and for infinite ones, look for limits for x→-∞ and x→+∞, respectively.

The task at this stage

With this service you can find the largest and smallest value of a function one variable f(x) with the solution formatted in Word. If the function f(x,y) is given, therefore, it is necessary to find the extremum of the function of two variables. You can also find the intervals of increasing and decreasing functions.

Find the largest and smallest value of a function

y =

on the segment [ ;]

Include theory

Rules for entering functions:

Necessary condition for the extremum of a function of one variable

The equation f" 0 (x *) = 0 is a necessary condition for the extremum of a function of one variable, i.e. at point x * the first derivative of the function must vanish. It identifies stationary points x c at which the function does not increase or decrease .

Sufficient condition for the extremum of a function of one variable

Let f 0 (x) be twice differentiable with respect to x belonging to the set D. If at point x * the condition is met:

F" 0 (x *) = 0
f"" 0 (x *) > 0

Then point x * is the local (global) minimum point of the function.

If at point x * the condition is met:

F" 0 (x *) = 0
f"" 0 (x *)< 0

Then point x * is a local (global) maximum.

Example No. 1. Find the largest and smallest values ​​of the function: on the segment.
Solution.

The critical point is one x 1 = 2 (f’(x)=0). This point belongs to the segment. (The point x=0 is not critical, since 0∉).
We calculate the values ​​of the function at the ends of the segment and at the critical point.
f(1)=9, f(2)= 5 / 2 , f(3)=3 8 / 81
Answer: f min = 5 / 2 at x=2; f max =9 at x=1

Example No. 2. Using higher order derivatives, find the extremum of the function y=x-2sin(x) .
Solution.
Find the derivative of the function: y’=1-2cos(x) . Let's find the critical points: 1-cos(x)=2, cos(x)=½, x=± π / 3 +2πk, k∈Z. We find y’’=2sin(x), calculate , which means x= π / 3 +2πk, k∈Z are the minimum points of the function; , which means x=- π / 3 +2πk, k∈Z are the maximum points of the function.

Example No. 3. Investigate the extremum function in the vicinity of the point x=0.
Solution. Here it is necessary to find the extrema of the function. If the extremum x=0, then find out its type (minimum or maximum). If among the found points there is no x = 0, then calculate the value of the function f(x=0).
It should be noted that when the derivative on each side of a given point does not change its sign, the possible situations are not exhausted even for differentiable functions: it can happen that for an arbitrarily small neighborhood on one side of the point x 0 or on both sides the derivative changes sign. At these points it is necessary to use other methods to study functions at an extremum.

Let the function y =f(X) is continuous on the interval [ a, b]. As is known, such a function reaches its maximum and minimum values ​​on this segment. The function can take these values ​​either at the internal point of the segment [ a, b], or on the boundary of the segment.

To find the largest and smallest values ​​of a function on the segment [ a, b] necessary:

1) find the critical points of the function in the interval ( a, b);

2) calculate the values ​​of the function at the found critical points;

3) calculate the values ​​of the function at the ends of the segment, that is, when x=A and x = b;

4) from all calculated values ​​of the function, select the largest and smallest.

Example. Find the largest and smallest values ​​of a function

on the segment.

Finding critical points:

These points lie inside the segment ; y(1) = ‒ 3; y(2) = ‒ 4; y(0) = ‒ 8; y(3) = 1;

at the point x= 3 and at the point x= 0.

Study of a function for convexity and inflection point.

Function y = f (x) called convexup in between (a, b) , if its graph lies under the tangent drawn at any point in this interval, and is called convex down (concave), if its graph lies above the tangent.

The point through which convexity is replaced by concavity or vice versa is called inflection point.

Algorithm for examining convexity and inflection point:

1. Find critical points of the second kind, that is, points at which the second derivative is equal to zero or does not exist.

2. Plot critical points on the number line, dividing it into intervals. Find the sign of the second derivative on each interval; if , then the function is convex upward, if, then the function is convex downward.

3. If, when passing through a critical point of the second kind, the sign changes and at this point the second derivative is equal to zero, then this point is the abscissa of the inflection point. Find its ordinate.

Asymptotes of the graph of a function. Study of a function for asymptotes.

Definition. The asymptote of the graph of a function is called straight, which has the property that the distance from any point on the graph to this line tends to zero as the point on the graph moves indefinitely from the origin.

There are three types of asymptotes: vertical, horizontal and inclined.

Definition. The straight line is called vertical asymptote function graphics y = f(x), if at least one of the one-sided limits of the function at this point is equal to infinity, that is

where is the discontinuity point of the function, that is, it does not belong to the domain of definition.

Example.

D ( y) = (‒ ∞; 2) (2; + ∞)

x= 2 – break point.

Definition. Straight y =A called horizontal asymptote function graphics y = f(x) at , if

Example.

x
y

Definition. Straight y =kx +b (k≠ 0) is called oblique asymptote function graphics y = f(x) at , where

General scheme for studying functions and constructing graphs.

Function Research Algorithmy = f(x) :

1. Find the domain of the function D (y).

2. Find (if possible) the points of intersection of the graph with the coordinate axes (if x= 0 and at y = 0).

3. Examine the evenness and oddness of the function ( y (‒ x) = y (x) ‒ parity; y(‒ x) = ‒ y (x) ‒ odd).

4. Find the asymptotes of the graph of the function.

5. Find the intervals of monotonicity of the function.

6. Find the extrema of the function.

7. Find the intervals of convexity (concavity) and inflection points of the function graph.

8. Based on the research conducted, construct a graph of the function.

Example. Explore the function and build its graph.

1) D (y) =

x= 4 – break point.

2) When x = 0,

(0; ‒ 5) – point of intersection with oh.

At y = 0,

3) y(‒ x)= a function of general form (neither even nor odd).

4) We examine for asymptotes.

a) vertical

b) horizontal

c) find the oblique asymptotes where

‒oblique asymptote equation

5) In this equation it is not necessary to find intervals of monotonicity of the function.

6)

These critical points divide the entire domain of definition of the function into the interval (˗∞; ˗2), (˗2; 4), (4; 10) and (10; +∞). It is convenient to present the results obtained in the form of the following table.