The largest integer value of the function. How to find the largest value of a function? Methodological recommendations for practical classes Topic: Introduction

In many areas of life, you may be faced with the need to solve something using numbers, for example, in economics and accounting, you can only find out the minimum and maximum of some indicators using optimization specified parameters. And this is nothing more than finding the largest and smallest values ​​on a given segment of the function. Now let's look at how to find highest value functions.

Finding the greatest value: instructions

1. Find out on which segment of the function you need to calculate the value, designate it with dots. This interval can be open (when the function is equal to the segment), closed (when the function is on the segment) and infinite (when the function does not end).
2. Find the derivative function.
3. Find the points on the function segment where the derivative is equal to zero, and that’s it critical points. Then calculate the values ​​of the function at these points and solve the equation. Find the largest among the obtained values.
4. Reveal function values ​​on end points, determine the greater of them
5. Compare the data with the largest value and select the largest one. This will be the largest value of the function.

How to find the largest integer value of a function? You need to calculate whether the function is even or odd, and then solve concrete example. If the number is obtained with a fraction, do not take it into account; the result of the largest integer value of the function will only be an integer.

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 behavior is constantly required to be assessed 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 by condition specific task it is necessary to determine the greatest 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 in the expression, square root etc. 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

Methodological recommendations for studying the topic “Set of function values. The largest and smallest values ​​of a function.”

In mathematics itself the main means

to achieve truth - induction and analogy.

Given: - function. Let's denote
- domain of definition of the function.

The set (domain) of values ​​of a function is the set of all those values ​​that a function can take
.Geometrically, this means the projection of the graph of a function onto the axis
.

If there is a point such that for anyone of the set there is an inequality
, then they say that the function on the set takes on its nai lower value

If there is a point such that for any of the set the inequality holds
, then they say that the function on the set takes on its highest value .

The function is called bounded below on the set if such a number exists
. Geometrically, this means that the graph of the function is not lower than the straight line
.

The function is called bounded above on the set if such a number exists , that for any of the set the inequality is true
. Geometrically, this means that the graph of the function is no higher than the straight line

The function is called limited on the set if it is bounded on this set from below and above. The boundedness of a function means that its graph is inside a certain horizontal band.

Cauchy's inequality about the arithmetic mean and the geometric mean
:

>,>0) Example:

The largest and smallest values ​​of a function on an interval

(segment, interval, ray)

Properties of functions continuous on an interval.

1. If a function is continuous on a segment, then it reaches both its maximum and its minimum values ​​on it.

2. A continuous function can reach its maximum and minimum values ​​both at the ends of a segment and inside it

3. If the largest (or smallest) value is achieved inside the segment, then only at a stationary or critical point.

Algorithm for finding the largest and smallest values continuous function on the segment

1. Find the derivative
.

2. Find stationary and critical points lying inside the segment .

3. Find the values ​​of the function at selected stationary and critical points and at the ends of the segment, i.e.
And
.

4.Among the found values, select the smallest (this will be
) and the greatest (this will be
)

Properties of continuous functions that are monotonic on an interval:

Continuous increasing on a segment the function reaches its greatest value at
, the smallest – at
.

Continuous decreasing on a segment the function reaches its greatest value at , and its minimum at .

If the function value
nonnegative on some interval, then this function and the function
, where n is a natural number, takes the largest (smallest) value at the same point.

Finding the largest and smallest values continuous function on the interval
or on beam

(optimization problems).

If a continuous function has a single extremum point on an interval or ray and this extremum is a maximum or minimum, then at this point the greatest or smallest value functions ( or )

Application of the property of monotonicity of functions.

1. A complex function composed of two increasing functions is increasing.

2.If the function increases and the function
decreases, then the function
- decreasing.

3. The sum of two increasing (decreasing) functions, increasing (decreasing) function.

4. If in Eq.
the left side is an increasing (or decreasing) function, then the equation has at most one root.

5.If the function is increasing (decreasing), and the function is decreasing (increasing), then the equation
has at most one solution.

6. Equation
has at least one root if and only if

belongs to multiple meanings
functions .

Application of the property of bounded functions.

1. If the left side of the equation (inequality) (
less than or equal to some number (
), A right side is greater than or equal to this number (), then the system holds
the solution of which is the solution to the equation (inequality) itself.

Self-control tasks

Application:

3. Find all values ​​for which the equation
has a solution.

Homework

1.Find the largest value of the function:

, If
.

2. Find the smallest value of the function:

.

3. Find the largest integer value of the function:

. those that correspond to greatest. Ideal-...

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 ​​explicitly given function one variable y=f(x) .

Page navigation.

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 a certain point, then this point is stationary. Thus, the function often takes its greatest (smallest) value on the interval X at one of the stationary points from this interval.

Also, a function can often take its greatest and minimum 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 stationary point, and the greatest - 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 straight line x=2 is 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 a function is the entire set real numbers, except for 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):

In this article I will talk about how to apply the skill of finding to the study of a function: to find its largest or smallest value. And then we will solve several problems from Task B15 from Open Bank tasks for .

As usual, let's first remember the theory.

At the beginning of any study of a function, we find it

To find the largest or smallest value of a function, you need to examine on which intervals the function increases and on which it decreases.

To do this, we need to find the derivative of the function and examine its intervals of constant sign, that is, the intervals over which the derivative retains its sign.

Intervals over which the derivative of a function is positive are intervals of increasing function.

Intervals over which the derivative of a function is negative are intervals of decreasing function.

1. Let's solve task B15 (No. 245184)

To solve it, we will follow the following algorithm:

a) Find the domain of definition of the function

b) Let's find the derivative of the function.

c) Let's equate it to zero.

d) Let us find the intervals of constant sign of the function.

e) Find the point at which the function takes on the greatest value.

f) Find the value of the function at this point.

I explain the detailed solution to this task in the VIDEO TUTORIAL:

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2. Let's solve task B15 (No. 282862)

Find the largest value of the function on the segment

It is obvious that the function takes the greatest value on the segment at the maximum point, at x=2. Let's find the value of the function at this point:

Answer: 5

3. Let's solve task B15 (No. 245180):

Find the largest value of the function

1. title="ln5>0">, , т.к. title="5>1">, поэтому это число не влияет на знак неравенства.!}

2. Because according to the domain of definition of the original function title="4-2x-x^2>0">, следовательно знаменатель дроби всегда больще нуля и дробь меняет знак только в нуле числителя.!}

3. Numerator equal to zero at . Let's check if it belongs ODZ functions. To do this, let’s check whether the condition title="4-2x-x^2>0"> при .!}

Title="4-2(-1)-((-1))^2>0">,

this means that the point belongs to the ODZ function

Let's examine the sign of the derivative to the right and left of the point:

We see that the function takes on its greatest value at point . Now let's find the value of the function at:

Remark 1. Note that in this problem we did not find the domain of definition of the function: we only fixed the restrictions and checked whether the point at which the derivative is equal to zero belongs to the domain of definition of the function. This turned out to be sufficient for this task. However, this is not always the case. It depends on the task.

Note 2. When studying behavior complex function you can use this rule:

• if the external function of a complex function is increasing, then the function takes its greatest value at the same point at which internal function takes the greatest value. This follows from the definition of an increasing function: a function increases on interval I if higher value the argument from this interval corresponds to a larger value of the function.
• if the outer function of a complex function is decreasing, then the function takes its greatest value at the same point at which the inner function takes its smallest value . This follows from the definition of a decreasing function: a function decreases on interval I if a larger value of the argument from this interval corresponds to a smaller value of the function

In our example, the external function increases throughout the entire domain of definition. Under the sign of the logarithm there is an expression - quadratic trinomial, which, with a negative leading coefficient, takes the greatest value at the point . Next, we substitute this x value into the function equation and find its greatest value.