How to find the smallest value of a function using its derivative. Using the derivative to find the largest and smallest values ​​of a continuous function on an interval

In practice, it is quite common to use the derivative in order to calculate the largest and smallest value of a function. We perform this action when we figure out how to minimize costs, increase profits, calculate the optimal load on production, etc., that is, in cases where we need to determine the optimal value of a parameter. To solve such problems correctly, you need to have a good understanding of what the largest and smallest values ​​of a function are.

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Typically we define these values ​​within a certain interval x, which in turn may correspond to the entire domain of the function or part of it. It can be like a segment [a; b ] , and open interval (a ; b), (a ; b ], [ a ; b), infinite interval (a ; b), (a ; b ], [ a ; b) or infinite interval - ∞ ; a , (- ∞ ; a ] , [ a ; + ∞) , (- ∞ ; + ∞) .

In this article we will tell you how to calculate the largest and smallest value explicitly given function with one variable y=f(x) y = f (x) .

Basic definitions

Let's start, as always, with the formulation of basic definitions.

Definition 1

The largest value of the function y = f (x) on a certain interval x is the value m a x y = f (x 0) x ∈ X, which for any value x x ∈ X, x ≠ x 0 makes the inequality f (x) ≤ f (x) valid 0) .

Definition 2

The smallest value of the function y = f (x) on a certain interval x is the value m i n x ∈ X y = f (x 0) , which for any value x ∈ X, x ≠ x 0 makes the inequality f(X f (x) ≥ f (x 0) .

These definitions are quite obvious. Even simpler, we can say this: the greatest value of a function is its most great value on a known interval at abscissa x 0, and the smallest is the smallest accepted value on the same interval at x 0.

Definition 3

Stationary points are those values ​​of the argument of a function at which its derivative becomes 0.

Why do we need to know what stationary points are? To answer this question, we need to remember Fermat's theorem. It follows from it that a stationary point is the point at which the extremum of the differentiable function is located (i.e., its local minimum or maximum). Consequently, the function will take the smallest or largest value on a certain interval precisely at one of the stationary points.

A function can also take on the largest or smallest value at those points at which the function itself is defined and its first derivative does not exist.

The first question that arises when studying this topic: in all cases can we determine the largest or smallest value of a function on a given interval? No, we cannot do this when the boundaries of a given interval coincide with the boundaries of the definition area, or if we are dealing with an infinite interval. It also happens that a function in a given segment or at infinity will take infinitely small or infinitely large values. In these cases, it is not possible to determine the largest and/or smallest value.

These points will become clearer after being depicted on the graphs:

The first figure shows us a function that takes the largest and smallest values ​​(m a x y and m i n y) at stationary points located on the segment [ - 6 ; 6].

Let us examine in detail the case indicated in the second graph. Let's change the value of the segment to [ 1 ; 6 ] and we find that the maximum value of the function will be achieved at the point with the abscissa at the right boundary of the interval, and the minimum - at the stationary point.

In the third figure, the abscissas of the points represent the boundary points of the segment [ - 3 ; 2]. They correspond to the largest and smallest value of a given function.

Now let's look at the fourth picture. In it, the function takes m a x y (the largest value) and m i n y (the smallest value) at stationary points on the open interval (- 6 ; 6).

If we take the interval [ 1 ; 6), then we can say that the smallest value of the function on it will be achieved at a stationary point. The greatest value will be unknown to us. The function could take its maximum value at x equal to 6 if x = 6 belonged to the interval. This is exactly the case shown in graph 5.

In graph 6, this function acquires its smallest value at the right boundary of the interval (- 3; 2 ], and we cannot draw definite conclusions about the largest value.

In Figure 7 we see that the function will have m a x y at a stationary point having an abscissa equal to 1. The function will reach its minimum value at the boundary of the interval c right side. At minus infinity, the function values ​​will asymptotically approach y = 3.

If we take the interval x ∈ 2; + ∞ , then we will see that the given function will take neither the smallest nor the largest value on it. If x tends to 2, then the values ​​of the function will tend to minus infinity, since the straight line x = 2 is a vertical asymptote. If the abscissa tends to plus infinity, then the function values ​​will asymptotically approach y = 3. This is exactly the case shown in Figure 8.

In this paragraph we will present the sequence of actions that need to be performed to find the largest or smallest value of a function on a certain segment.

1. First, let's find the domain of definition of the function. Let's check whether the segment specified in the condition is included in it.
2. Now let's calculate the points contained in this segment at which the first derivative does not exist. Most often they can be found in functions whose argument is written under the modulus sign, or in power functions, the exponent of which is a fractionally rational number.
3. Next, we will find out which stationary points will fall in the given segment. To do this, you need to calculate the derivative of the function, then equate it to 0 and solve the resulting equation, and then select the appropriate roots. If we don’t get a single stationary point or they don’t fall into the given segment, then we move on to the next step.
4. We determine what values ​​the function will take at given stationary points (if any), or at those points at which the first derivative does not exist (if there are any), or we calculate the values ​​for x = a and x = b.
5. 5. We have a number of function values, from which we now need to select the largest and smallest. These will be the largest and smallest values ​​of the function that we need to find.

Let's see how to correctly apply this algorithm when solving problems.

Example 1

Condition: the function y = x 3 + 4 x 2 is given. Determine its largest and smallest values ​​on the segments [ 1 ; 4 ] and [ - 4 ; - 1 ] .

Solution:

Let's start by finding the domain of definition of a given function. In this case, she will have a lot of everyone real numbers, except 0 . In other words, D (y) : x ∈ (- ∞ ; 0) ∪ 0 ; + ∞ . Both segments specified in the condition will be inside the definition area.

Now we calculate the derivative of the function according to the rule of fraction differentiation:

y " = x 3 + 4 x 2 " = x 3 + 4 " x 2 - x 3 + 4 x 2 " x 4 = = 3 x 2 x 2 - (x 3 - 4) 2 x x 4 = x 3 - 8 x 3

We learned that the derivative of a function will exist at all points of the segments [ 1 ; 4 ] and [ - 4 ; - 1 ] .

Now we need to determine the stationary points of the function. Let's do this using the equation x 3 - 8 x 3 = 0. It has only one real root, which is 2. It will be a stationary point of the function and will fall into the first segment [1; 4].

Let us calculate the values ​​of the function at the ends of the first segment and at this point, i.e. for x = 1, x = 2 and x = 4:

y (1) = 1 3 + 4 1 2 = 5 y (2) = 2 3 + 4 2 2 = 3 y (4) = 4 3 + 4 4 2 = 4 1 4

We found that the largest value of the function m a x y x ∈ [ 1 ; 4 ] = y (2) = 3 will be achieved at x = 1, and the smallest m i n y x ∈ [ 1 ; 4 ] = y (2) = 3 – at x = 2.

The second segment does not include a single stationary point, so we need to calculate the function values ​​only at the ends of the given segment:

y (- 1) = (- 1) 3 + 4 (- 1) 2 = 3

This means m a x y x ∈ [ - 4 ; - 1 ] = y (- 1) = 3 , m i n y x ∈ [ - 4 ; - 1 ] = y (- 4) = - 3 3 4 .

Answer: For the segment [ 1 ; 4 ] - m a x y x ∈ [ 1 ; 4 ] = y (2) = 3 , m i n y x ∈ [ 1 ; 4 ] = y (2) = 3 , for the segment [ - 4 ; - 1 ] - m a x y x ∈ [ - 4 ; - 1 ] = y (- 1) = 3 , m i n y x ∈ [ - 4 ; - 1 ] = y (- 4) = - 3 3 4 .

See picture:

Before studying this method, we advise you to review how to correctly calculate the one-sided limit and the limit at infinity, as well as learn the basic methods for finding them. To find the largest and/or smallest value of a function on an open or infinite interval, perform the following steps sequentially.

1. First, you need to check whether the given interval will be a subset of the domain of the given function.
2. Let us determine all points that are contained in the required interval and at which the first derivative does not exist. They usually occur in functions where the argument is enclosed in the modulus sign, and in power functions with fractional rational indicator. If these points are missing, then you can proceed to the next step.
3. Now let’s determine which stationary points will fall within the given interval. First, we equate the derivative to 0, solve the equation and select suitable roots. If we do not have a single stationary point or they do not fall within the given interval, then we immediately go to further actions. They are determined by the type of interval.

• If the interval is of the form [ a ; b) , then we need to calculate the value of the function at the point x = a and the one-sided limit lim x → b - 0 f (x) .
• If the interval has the form (a; b ], then we need to calculate the value of the function at the point x = b and the one-sided limit lim x → a + 0 f (x).
• If the interval has the form (a; b), then we need to calculate the one-sided limits lim x → b - 0 f (x), lim x → a + 0 f (x).
• If the interval is of the form [ a ; + ∞), then we need to calculate the value at the point x = a and the limit at plus infinity lim x → + ∞ f (x) .
• If the interval looks like (- ∞ ; b ] , we calculate the value at the point x = b and the limit at minus infinity lim x → - ∞ f (x) .
• If - ∞ ; b , then we consider the one-sided limit lim x → b - 0 f (x) and the limit at minus infinity lim x → - ∞ f (x)
• If - ∞; + ∞ , then we consider the limits on minus and plus infinity lim x → + ∞ f (x) , lim x → - ∞ f (x).

1. At the end, you need to draw a conclusion based on the obtained function values ​​and limits. There are many options available here. So, if the one-sided limit is equal to minus infinity or plus infinity, then it is immediately clear that nothing can be said about the smallest and largest values ​​of the function. Below we will look at one typical example. Detailed Descriptions will help you understand what's what. If necessary, you can return to Figures 4 - 8 in the first part of the material.

Condition: given function y = 3 e 1 x 2 + x - 6 - 4 . Calculate its largest and smallest value in the intervals - ∞ ; - 4, - ∞; - 3 , (- 3 ; 1 ] , (- 3 ; 2) , [ 1 ; 2) , 2 ; + ∞ , [ 4 ; + ∞) .

Solution

First of all, we find the domain of definition of the function. The denominator of the fraction contains a quadratic trinomial, which should not turn to 0:

x 2 + x - 6 = 0 D = 1 2 - 4 1 (- 6) = 25 x 1 = - 1 - 5 2 = - 3 x 2 = - 1 + 5 2 = 2 ⇒ D (y) : x ∈ (- ∞ ; - 3) ∪ (- 3 ; 2) ∪ (2 ; + ∞)

We have obtained the domain of definition of the function to which all the intervals specified in the condition belong.

Now let's differentiate the function and get:

y" = 3 e 1 x 2 + x - 6 - 4 " = 3 e 1 x 2 + x - 6 " = 3 e 1 x 2 + x - 6 1 x 2 + x - 6 " = = 3 · e 1 x 2 + x - 6 · 1 " · x 2 + x - 6 - 1 · x 2 + x - 6 " (x 2 + x - 6) 2 = - 3 · (2 ​​x + 1) · e 1 x 2 + x - 6 x 2 + x - 6 2

Consequently, derivatives of a function exist throughout its entire domain of definition.

Let's move on to finding stationary points. The derivative of the function becomes 0 at x = - 1 2 . This is a stationary point that lies in the intervals (- 3 ; 1 ] and (- 3 ; 2) .

Let's calculate the value of the function at x = - 4 for the interval (- ∞ ; - 4 ], as well as the limit at minus infinity:

y (- 4) = 3 e 1 (- 4) 2 + (- 4) - 6 - 4 = 3 e 1 6 - 4 ≈ - 0 . 456 lim x → - ∞ 3 e 1 x 2 + x - 6 = 3 e 0 - 4 = - 1

Since 3 e 1 6 - 4 > - 1, it means that m a x y x ∈ (- ∞ ; - 4 ] = y (- 4) = 3 e 1 6 - 4. This does not allow us to uniquely determine the smallest value of the function. We can only conclude that there is a constraint below - 1, since it is to this value that the function approaches asymptotically at minus infinity.

The peculiarity of the second interval is that there is not a single stationary point and not a single strict boundary in it. Consequently, we will not be able to calculate either the largest or smallest value of the function. Having defined the limit at minus infinity and as the argument tends to - 3 on the left side, we get only an interval of values:

lim x → - 3 - 0 3 e 1 x 2 + x - 6 - 4 = lim x → - 3 - 0 3 e 1 (x + 3) (x - 3) - 4 = 3 e 1 (- 3 - 0 + 3) (- 3 - 0 - 2) - 4 = = 3 e 1 (+ 0) - 4 = 3 e + ∞ - 4 = + ∞ lim x → - ∞ 3 e 1 x 2 + x - 6 - 4 = 3 e 0 - 4 = - 1

This means that the function values ​​will be located in the interval - 1; +∞

To find the largest value of the function in the third interval, we determine its value at the stationary point x = - 1 2 if x = 1. We will also need to know the one-sided limit for the case when the argument tends to - 3 on the right side:

y - 1 2 = 3 e 1 - 1 2 2 + - 1 2 - 6 - 4 = 3 e 4 25 - 4 ≈ - 1 . 444 y (1) = 3 e 1 1 2 + 1 - 6 - 4 ≈ - 1 . 644 lim x → - 3 + 0 3 e 1 x 2 + x - 6 - 4 = lim x → - 3 + 0 3 e 1 (x + 3) (x - 2) - 4 = 3 e 1 - 3 + 0 + 3 (- 3 + 0 - 2) - 4 = = 3 e 1 (- 0) - 4 = 3 e - ∞ - 4 = 3 0 - 4 = - 4

It turned out that the function will take the greatest value at a stationary point m a x y x ∈ (3; 1 ] = y - 1 2 = 3 e - 4 25 - 4. As for the smallest value, we cannot determine it. Everything we know , is the presence of a lower limit to - 4 .

For the interval (- 3 ; 2), take the results of the previous calculation and once again calculate what the one-sided limit is equal to when tending to 2 on the left side:

y - 1 2 = 3 e 1 - 1 2 2 + - 1 2 - 6 - 4 = 3 e - 4 25 - 4 ≈ - 1 . 444 lim x → - 3 + 0 3 e 1 x 2 + x - 6 - 4 = - 4 lim x → 2 - 0 3 e 1 x 2 + x - 6 - 4 = lim x → - 3 + 0 3 e 1 (x + 3) (x - 2) - 4 = 3 e 1 (2 - 0 + 3) (2 - 0 - 2) - 4 = = 3 e 1 - 0 - 4 = 3 e - ∞ - 4 = 3 · 0 - 4 = - 4

This means that m a x y x ∈ (- 3 ; 2) = y - 1 2 = 3 e - 4 25 - 4, and the smallest value cannot be determined, and the values ​​of the function are limited from below by the number - 4.

Based on what we got in the two previous calculations, we can say that on the interval [ 1 ; 2) the function will take the largest value at x = 1, but it is impossible to find the smallest.

On the interval (2 ; + ∞) the function will not reach either the largest or the smallest value, i.e. it will take values ​​from the interval - 1 ; + ∞ .

lim x → 2 + 0 3 e 1 x 2 + x - 6 - 4 = lim x → - 3 + 0 3 e 1 (x + 3) (x - 2) - 4 = 3 e 1 (2 + 0 + 3 ) (2 + 0 - 2) - 4 = = 3 e 1 (+ 0) - 4 = 3 e + ∞ - 4 = + ∞ lim x → + ∞ 3 e 1 x 2 + x - 6 - 4 = 3 e 0 - 4 = - 1

Having calculated what the value of the function will be equal to at x = 4, we find out that m a x y x ∈ [ 4 ; + ∞) = y (4) = 3 e 1 14 - 4 , and the given function at plus infinity will asymptotically approach the straight line y = - 1 .

Let's compare what we got in each calculation with the graph of the given function. In the figure, the asymptotes are shown by dotted lines.

That's all we wanted to tell you about finding the largest and smallest values ​​of a function. The sequences of actions that we have given will help you make the necessary calculations as quickly and simply as possible. But remember that it is often useful to first find out at which intervals the function will decrease and at which it will increase, after which you can draw further conclusions. This way you can more accurately determine the largest and smallest values ​​of the function and justify the results obtained.

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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 specified 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 greatest (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):

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.

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)= function general view(neither even nor odd).

4) We examine for asymptotes.

a) vertical

b) horizontal

c) find the oblique asymptotes where

‒oblique asymptote equation

5) B given equation there is no need 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.

Hello! In this article we will talk about problems that can be solved without finding the derivative. In this section we have already looked at some examples.The meaning of the tasks is the same - you need to find either the maximum (minimum) point of the function, or determine the maximum (minimum) value of the function.

What is the essence and what is the “standard” solution algorithm - you can see in. But the use of this algorithm will not be rational for all tasks. If you follow it in the examples below, the solution process will be “overloaded” with calculations. You don't need to waste time on the exam.So what tasks do we mean?

The condition gives an irrational, logarithmic or exponential function:

where under the root, under the sign of the logarithm or in the exponent there is a quadratic function of the form:

Let's consider the approach without finding the derivative. You will see that such problems can be solved orally.

What do you need to know?The property of a parabola, let us recall it:

If a > 0, then its branches are directed upward.

If a< 0, то её ветви направлены вниз.

That is, this is the extremum point quadratic function– in it the function changes its behavior from increasing to decreasing or vice versa.

Next important fact(key for these tasks):

If the original function is monotonic (continuously increasing or decreasing), for it the specified point “x” will also be an extremum point.

Why? Let's look at the functions separately in more detail.

Quadratic function in exponent (with n>1):

Look!

It turns out that the value of z changes as follows.

Option when a>0 (the branches of the parabola are directed upward) – at x from minus infinity to –b/2a z decreases, at the point –b/2a the value will be minimal, then at x from –b/2a to infinity z increases.

This means that the function y=n f (x) itself will have a minimum value at the point x=–b/2a, since a minimum in the indicator will result in a minimum in the result.

Option when a<0 (ветви параболы направлены вниз) – при х от минус бесконечности до –b/2a z увеличивается, в точке –b/2a значение будет максимальным, далее при х от –b/2a до бесконечности z уменьшается.

This means that the function y=n f (x) itself will have a maximum value at the point x=–b/2a, since with a maximum in the indicator there will be a maximum in the result.

Quadratic function under the logarithm sign (with n>1):

Let's imagine that ax 2 +bx+c=z. We can write:

It turns out that the value of z changes as follows:

Option when a>0 (the branches of the parabola are directed upward) – at x from minus infinity to –b/2a z decreases, at the point –b/2a the value will be minimal, then at x from –b/2a to infinity z increases.

This means that the log n z function itself will have a minimum value at the point x=–b/2a. Since the logarithmic function decreases as the argument decreases (can be seen from the graph).

Option when a<0 (ветви параболы направлены вниз) – при х от минус бесконечности до –b/2a z увеличивается, в точке –b/2a значение будет максимальным, далее при х от –b/2a до бесконечности z уменьшается.

This means that the log n z function itself will have its maximum value at the point x=–b/2a. Since the logarithmic function increases as the argument increases (can be seen from the graph).

Quadratic function under the root sign:

Let's imagine that ax 2 +bx+c=z. We can write:

It turns out that:

When a>0, the value of z is minimal at the point x=–b/2a, which means the function itself will have a minimal value. *The root of the smallest value will result in the smallest number.

When a<0 значение z максимально в точке х=–b/2a, а значит и сама функция будет иметь максимальное значение.

Thus, let us formulate the key rule:

ATTENTION! Of course, if you go deeper into the topic, then options are possible when a complex function has a negative sign, when the logarithm is in the denominator of the fraction, when the base of the logarithm or the base of the power is in the range from 0 to 1. Of course, it is important to understand how the function given in the condition behaves (increases or decreases). But to solve typical exam tasks, the indicated conclusion will be quite enough for you.

And of course, do not lose sight of the range of acceptable values ​​of a given function:

— an expression under the root sign, greater than or equal to zero (a non-negative number).

- the expression under the logarithm sign is a positive number.

- the expression in the denominator of the fraction is not equal to zero.

In similar problems of finding the largest and smallest values ​​of a function, I would advise finding the domain of definition in any case (even though in the examples presented below this does not give us anything important and does not affect the answer).

Let's look at examples:

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How to find the largest and smallest values ​​of a function on a segment?

For this we follow a well-known algorithm:

1 . Finding the ODZ functions.

2 . Finding the derivative of the function

3 . Equating the derivative to zero

4 . We find the intervals over which the derivative retains its sign, and from them we determine the intervals of increase and decrease of the function:

If on interval I the derivative of the function is 0" title="f^(prime)(x)>0">, то функция !} increases over this interval.

If on the interval I the derivative of the function , then the function decreases over this interval.

5 . We find maximum and minimum points of the function.

IN at the maximum point of the function, the derivative changes sign from “+” to “-”.

IN minimum point of the functionthe derivative changes sign from "-" to "+".

6 . We find the value of the function at the ends of the segment,

• then we compare the value of the function at the ends of the segment and at the maximum points, and choose the largest of them if you need to find the largest value of the function
• or compare the value of the function at the ends of the segment and at the minimum points, and choose the smallest of them if you need to find the smallest value of the function

However, depending on how the function behaves on the segment, this algorithm can be significantly reduced.

Consider the function . The graph of this function looks like this:

Let's look at a few examples of solving problems from Open Bank tasks for

1. Task B15 (No. 26695)

On the segment.

1. The function is defined for all real values ​​of x

Obviously, this equation has no solutions, and the derivative is positive for all values ​​of x. Consequently, the function increases and takes on the greatest value at the right end of the interval, that is, at x=0.

Answer: 5.

2 . Task B15 (No. 26702)

Find the largest value of the function on the segment.

1. ODZ functions title="x(pi)/2+(pi)k, k(in)(bbZ)">!}

The derivative is equal to zero at , however, at these points it does not change sign:

Therefore, title="3/(cos^2(x))>=3">, значит, title="3/(cos^2(x))-3>=0">, то есть производная при всех допустимых значених х неотрицательна, следовательно, функция !} increases and takes the greatest value at the right end of the interval, at .

To make it obvious why the derivative does not change sign, we transform the expression for the derivative as follows:

Title="y^(prime)=3/(cos^2(x))-3=(3-3cos^2(x))/(cos^2(x))=(3sin^2 (x))/(cos^2(x))=3tg^2(x)>=0">!}

Answer: 5.

3. Task B15 (No. 26708)

Find the smallest value of the function on the segment.

1. ODZ functions: title="x(pi)/2+(pi)k, k(in)(bbZ)">!}

Let's place the roots of this equation on the trigonometric circle.

The interval contains two numbers: and

Let's put up signs. To do this, we determine the sign of the derivative at the point x=0: . When passing through points and, the derivative changes sign.

Let us depict the change of signs of the derivative of a function on the coordinate line:

Obviously, the point is a minimum point (at which the derivative changes sign from “-” to “+”), and to find the smallest value of the function on the segment, you need to compare the values ​​of the function at the minimum point and at the left end of the segment, .