How to find extremum points of a function examples. What are extrema of a function: critical points of maximum and minimum

By using of this service 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.

Rules for entering functions:

Necessary condition for the extremum of a function of one variable

Sufficient condition for the extremum of a function of one variable

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

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

Definitions:

Extremum called the maximum or minimum value functions on a given set.

Extremum point is the point at which the maximum or minimum value of the function is reached.

Maximum point is the point at which the maximum value of the function is reached.

Minimum point is the point at which the minimum value of the function is reached.

Explanation.

In the figure, in the vicinity of the point x = 3, the function reaches its maximum value (that is, in the vicinity of this particular point there is no point higher). In the neighborhood of x = 8, it again has a maximum value (let us clarify again: it is in this neighborhood that there is no point higher). At these points, the increase gives way to a decrease. They are the maximum points:

x max = 3, x max = 8.

In the vicinity of the point x = 5, the minimum value of the function is reached (that is, in the vicinity of x = 5 there is no point below). At this point the decrease gives way to an increase. It is the minimum point:

The maximum and minimum points are extremum points of the function, and the values ​​of the function at these points are its extremes.

Critical and stationary points of the function:

Necessary condition for an extremum:

Sufficient condition for an extremum:

On a segment the function y = f(x) can reach its minimum or maximum value either at critical points or at the ends of the segment .

Algorithm for studying a continuous functiony = f(x) for monotonicity and extrema:

Increasing, decreasing and extrema of a function

Finding intervals of increasing, decreasing and extrema of a function is as follows: an independent task, so the most important part other tasks, in particular full function study. Initial information about the increase, decrease and extrema of the function is given in theoretical chapter on derivative, which I highly recommend for preliminary study (or repetition)– also for the reason that the following material is based on the very essentially derivative, being a harmonious continuation of this article. Although, if time is short, then a purely formal practice of examples from today’s lesson is also possible.

And today there is a spirit of rare unanimity in the air, and I can directly feel that everyone present is burning with desire learn to explore a function using its derivative. Therefore, reasonable, good, eternal terminology immediately appears on your monitor screens.

For what? One of the reasons is the most practical: so that it is clear what is generally required of you in a particular task!

Monotonicity of the function. Extremum points and extrema of a function

Let's consider some function. To put it simply, we assume that she continuous on the entire number line:

Just in case, let’s immediately get rid of possible illusions, especially for those readers who have recently become acquainted with intervals of constant sign of the function. Now we NOT INTERESTED, how the graph of the function is located relative to the axis (above, below, where the axis intersects). To be convincing, mentally erase the axes and leave one graph. Because that’s where the interest lies.

Function increases on an interval if for any two points of this interval connected by the relation , the inequality is true. That is, a larger value of the argument corresponds to a larger value of the function, and its graph goes “from bottom to top”. The demonstration function grows over the interval.

Likewise, the function decreases on an interval if for any two points of a given interval such that , the inequality is true. That is, a larger value of the argument corresponds to a smaller value of the function, and its graph goes “from top to bottom.” Our function decreases on intervals .

If a function increases or decreases over an interval, then it is called strictly monotonous at this interval. What is monotony? Understand in literally– monotony.

You can also define non-decreasing function (relaxed condition in the first definition) and non-increasing function (softened condition in the 2nd definition). A non-decreasing or non-increasing function on an interval is called a monotonic function on a given interval (strict monotony - special case“just” monotony).

The theory also considers other approaches to determining the increase/decrease of a function, including on half-intervals, segments, but in order not to pour oil-oil-oil on your head, we will agree to operate with open intervals with categorical definitions - this is clearer, and for solving many practical problems quite enough.

Thus, in my articles the wording “monotonicity of a function” will almost always be hidden intervals strict monotony(strictly increasing or strictly decreasing function).

Neighborhood of a point. Words after which students run away wherever they can and hide in horror in the corners. ...Although after the post Cauchy limits They’re probably no longer hiding, but just shuddering slightly =) Don’t worry, now there will be no proofs of the theorems of mathematical analysis - I needed the surroundings to formulate the definitions more strictly extremum points. Let's remember:

Neighborhood of a point called the interval that contains this point, while for convenience the interval is often assumed to be symmetrical. For example, a point and its standard neighborhood:

Actually, the definitions:

The point is called strict maximum point, If exists her neighborhood, for all values ​​of which, except for the point itself, the inequality . In our specific example, this is a dot.

The point is called strict minimum point, If exists her neighborhood, for all values ​​of which, except for the point itself, the inequality . In the drawing there is point “a”.

Note : the requirement of neighborhood symmetry is not at all necessary. In addition, it is important the very fact of existence neighborhood (whether tiny or microscopic) that satisfies the specified conditions

The points are called strictly extremum points or simply extremum points functions. That is, it is a generalized term for maximum points and minimum points.

How do we understand the word “extreme”? Yes, just as directly as monotony. Extreme points of roller coasters.

As in the case of monotonicity, loose postulates exist and are even more common in theory (which, of course, the strict cases considered fall under!):

The point is called maximum point, If exists its surroundings are such that for all
The point is called minimum point, If exists its surroundings are such that for all values ​​of this neighborhood, the inequality holds.

Note that according to the last two definitions, any point of a constant function (or a “flat section” of a function) is considered both a maximum and a minimum point! The function, by the way, is both non-increasing and non-decreasing, that is, monotonic. However, we will leave these considerations to theorists, since in practice we almost always contemplate traditional “hills” and “hollows” (see drawing) with a unique “king of the hill” or “princess of the swamp”. As a variety, it occurs tip, directed up or down, for example, the minimum of the function at the point.

Yes, by the way, oh royalty:
– the meaning is called maximum functions;
– the meaning is called minimum functions.

Common name – extremes functions.

Please be careful with your words!

Extremum points– these are “X” values.
Extremes– “game” meanings.

! Note : sometimes the listed terms refer to the “X-Y” points that lie directly on the GRAPH OF the function ITSELF.

How many extrema can a function have?

None, 1, 2, 3, ... etc. to infinity. For example, sine has infinitely many minima and maxima.

IMPORTANT! The term "maximum of function" not identical the term “maximum value of a function”. It is easy to notice that the value is maximum only in a local neighborhood, and at the top left there are “cooler comrades”. Likewise, “minimum of a function” is not the same as “minimum value of a function,” and in the drawing we see that the value is minimum only in a certain area. In this regard, extremum points are also called local extremum points, and the extrema – local extremes. They walk and wander nearby and global brethren. So, any parabola has at its vertex global minimum or global maximum. Further, I will not distinguish between types of extremes, and the explanation is voiced more for general educational purposes - the additional adjectives “local”/“global” should not take you by surprise.

Let’s summarize our short excursion into the theory with a test shot: what does the task “find the monotonicity intervals and extremum points of the function” mean?

The wording encourages you to find:

– intervals of increasing/decreasing function (non-decreasing, non-increasing appears much less often);

– maximum and/or minimum points (if any exist). Well, to avoid failure, it’s better to find the minimums/maximums themselves ;-)

How to determine all this? Using the derivative function!

How to find intervals of increasing, decreasing,
extremum points and extrema of the function?

Many rules, in fact, are already known and understood from lesson about the meaning of a derivative.

Tangent derivative brings the cheerful news that function is increasing throughout domain of definition.

With cotangent and its derivative the situation is exactly the opposite.

The arcsine increases over the interval - the derivative here is positive: .
When the function is defined, but not differentiable. However, at the critical point there is a right-handed derivative and a right-handed tangent, and at the other edge there are their left-handed counterparts.

I think it won’t be too difficult for you to carry out similar reasoning for the arc cosine and its derivative.

All of the above cases, many of which are tabular derivatives, I remind you, follow directly from derivative definitions.

Why explore a function using its derivative?

To better understand what the graph of this function looks like: where it goes “bottom up”, where “top down”, where it reaches minimums and maximums (if it reaches at all). Not all functions are so simple - in most cases we have no idea at all about the graph of a particular function.

It's time to move on to more meaningful examples and consider algorithm for finding intervals of monotonicity and extrema of a function:

Example 1

Find intervals of increase/decrease and extrema of the function

Solution:

1) The first step is to find domain of a function, and also take note of the break points (if they exist). In this case, the function is continuous on the entire number line, and this action is to a certain extent formal. But in a number of cases, serious passions flare up here, so let’s treat the paragraph without disdain.

2) The second point of the algorithm is due to

a necessary condition for an extremum:

If there is an extremum at a point, then either the value does not exist.

Confused by the ending? Extremum of the “modulus x” function .

The condition is necessary, but not enough, and the converse is not always true. So, it does not yet follow from the equality that the function reaches a maximum or minimum at point . A classic example has already been highlighted above - this is a cubic parabola and its critical point.

But be that as it may, the necessary condition for an extremum dictates the need to find suspicious points. To do this, find the derivative and solve the equation:

At the beginning of the first article about function graphs I told you how to quickly build a parabola using an example : “...we take the first derivative and equate it to zero: ...So, the solution to our equation: - it is at this point that the vertex of the parabola is located...”. Now, I think, everyone understands why the vertex of the parabola is located exactly at this point =) In general, we should start with a similar example here, but it is too simple (even for a teapot). In addition, there is an analogue at the very end of the lesson about derivative of the function. Therefore, let's increase the degree:

Example 2

Find intervals of monotonicity and extrema of the function

This is an example for independent decision. Complete solution and an approximate final sample of the task at the end of the lesson.

The long-awaited moment of meeting with fractional-rational functions has arrived:

Example 3

Explore a function using the first derivative

Pay attention to how variably one and the same task can be reformulated.

Solution:

1) The function suffers infinite discontinuities at points.

2) We detect critical points. Let's find the first derivative and equate it to zero:

Let's solve the equation. A fraction is equal to zero when its numerator equal to zero:

Thus, we get three critical points:

3) We plot ALL detected points on the number line and interval method we define the signs of the DERIVATIVE:

I remind you that you need to take some point in the interval and calculate the value of the derivative at it and determine its sign. It’s more profitable not to even count, but to “estimate” verbally. Let's take, for example, a point belonging to the interval and perform the substitution: .

Two “pluses” and one “minus” give a “minus”, therefore, which means that the derivative is negative over the entire interval.

The action, as you understand, needs to be carried out for each of the six intervals. By the way, note that the numerator factor and denominator are strictly positive for any point in any interval, which greatly simplifies the task.

So, the derivative told us that the FUNCTION ITSELF increases by and decreases by . It is convenient to connect intervals of the same type with the join icon.

At the point the function reaches its maximum:
At the point the function reaches a minimum:

Think about why you don't have to recalculate the second value ;-)

When passing through a point, the derivative does not change sign, so the function has NO EXTREMUM there - it both decreased and remained decreasing.

! Let's repeat important point : points are not considered critical - they contain a function not determined. Accordingly, here In principle there can be no extremes(even if the derivative changes sign).

Answer: function increases by and decreases by At the point the maximum of the function is reached: , and at the point – the minimum: .

Knowledge of monotonicity intervals and extrema, coupled with established asymptotes already gives a very good idea of appearance function graphics. A person of average training is able to verbally determine that the graph of a function has two vertical asymptotes and an oblique asymptote. Here is our hero:

Try once again to correlate the results of the study with the graph of this function.
There is no extremum at the critical point, but there is inflection point(which, as a rule, happens in similar cases).

Example 4

Find the extrema of the function

Example 5

Find monotonicity intervals, maxima and minima of the function

…it’s almost like some kind of “X in a cube” holiday today....
Soooo, who in the gallery offered to drink for this? =)

Each task has its own substantive nuances and technical details, which are commented out at the end of the lesson.

The extremum point of a function is the point in the domain of definition of the function at which the value of the function takes on a minimum or maximum value. The values ​​of the function at these points are called extrema (minimum and maximum) of the function.

Definition. Dot x1 function domain f(x) is called maximum point of the function , if the value of the function at this point is greater than the values ​​of the function at points sufficiently close to it, located to the right and left of it (that is, the inequality holds f(x0 ) > f(x 0 + Δ x) x1 maximum.

Definition. Dot x2 function domain f(x) is called minimum point of the function, if the value of the function at this point is less than the values ​​of the function at points sufficiently close to it, located to the right and left of it (that is, the inequality holds f(x0 ) < f(x 0 + Δ x) ). In this case we say that the function has at the point x2 minimum.

Let's say point x1 - maximum point of the function f(x) . Then in the interval up to x1 function increases, therefore the derivative of the function is greater than zero ( f "(x) > 0 ), and in the interval after x1 the function decreases, therefore, derivative of a function less than zero (f "(x) < 0 ). Тогда в точке x1

Let us also assume that the point x2 - minimum point of the function f(x) . Then in the interval up to x2 the function is decreasing, and the derivative of the function is less than zero ( f "(x) < 0 ), а в интервале после x2 the function is increasing, and the derivative of the function is greater than zero ( f "(x) > 0 ). In this case also at the point x2 the derivative of the function is zero or does not exist.

Fermat's theorem (a necessary sign of the existence of an extremum of a function). If the point x0 - extremum point of the function f(x) then at this point the derivative of the function is equal to zero ( f "(x) = 0 ) or does not exist.

Definition. The points at which the derivative of a function is zero or does not exist are called critical points .

Example 1. Let's consider the function.

At the point x= 0 the derivative of the function is zero, therefore the point x= 0 is the critical point. However, as can be seen in the graph of the function, it increases throughout the entire domain of definition, so the point x= 0 is not the extremum point of this function.

Thus, the conditions that the derivative of a function at a point is equal to zero or does not exist are necessary conditions for an extremum, but not sufficient, since other examples of functions can be given for which these conditions are met, but the function does not have an extremum at the corresponding point. That's why there must be sufficient evidence, allowing one to judge whether there is an extremum at a particular critical point and what kind of extremum it is - maximum or minimum.

Theorem (the first sufficient sign of the existence of an extremum of a function). Critical point x0 f(x) if, when passing through this point, the derivative of the function changes sign, and if the sign changes from “plus” to “minus”, then it is a maximum point, and if from “minus” to “plus”, then it is a minimum point.

If near the point x0 , to the left and to the right of it, the derivative retains its sign, this means that the function either only decreases or only increases in a certain neighborhood of the point x0 . In this case, at the point x0 there is no extreme.

So, to determine the extremum points of the function, you need to do the following :

1. Find the derivative of the function.
2. Equate the derivative to zero and determine the critical points.
3. Mentally or on paper, mark the critical points on the number line and determine the signs of the derivative of the function in the resulting intervals. If the sign of the derivative changes from “plus” to “minus”, then the critical point is the maximum point, and if from “minus” to “plus”, then the minimum point.
4. Calculate the value of the function at the extremum points.

Example 2. Find the extrema of the function .

Solution. Let's find the derivative of the function:

Let's equate the derivative to zero to find the critical points:

.

Since for any values ​​of “x” the denominator is not equal to zero, we equate the numerator to zero:

Got one critical point x= 3 . Let us determine the sign of the derivative in the intervals delimited by this point:

in the range from minus infinity to 3 - a minus sign, that is, the function decreases,

in the interval from 3 to plus infinity there is a plus sign, that is, the function increases.

That is, period x= 3 is the minimum point.

Let's find the value of the function at the minimum point:

Thus, the extremum point of the function is found: (3; 0), and it is the minimum point.

Theorem (the second sufficient sign of the existence of an extremum of a function). Critical point x0 is the extremum point of the function f(x) if the second derivative of the function at this point is not equal to zero ( f ""(x) ≠ 0 ), and if the second derivative is greater than zero ( f ""(x) > 0 ), then the maximum point, and if the second derivative is less than zero ( f ""(x) < 0 ), то точкой минимума.

Note 1. If at the point x0 If both the first and second derivatives vanish, then at this point it is impossible to judge the presence of an extremum based on the second sufficient criterion. In this case, you need to use the first sufficient criterion for the extremum of a function.

Remark 2. The second sufficient criterion for the extremum of a function is not applicable even when the first derivative does not exist at a stationary point (then the second derivative does not exist either). In this case, you also need to use the first sufficient sign of an extremum of a function.

Local nature of the extrema of the function

From the above definitions it follows that the extremum of a function is local in nature - it is the largest and smallest value of the function compared to nearby values.

Let's say you're looking at your earnings over a period of one year. If in May you earned 45,000 rubles, and in April 42,000 rubles and in June 39,000 rubles, then May earnings are the maximum of the earnings function compared to nearby values. But in October you earned 71,000 rubles, in September 75,000 rubles, and in November 74,000 rubles, so October earnings are the minimum of the earnings function compared to nearby values. And you can easily see that the maximum among the values ​​of April-May-June is less than the minimum of September-October-November.

Generally speaking, on an interval a function can have several extrema, and it may turn out that some minimum of the function is greater than any maximum. So, for the function shown in the figure above, .

That is, one should not think that the maximum and minimum of a function are, respectively, its largest and smallest values ​​on the entire segment under consideration. At the maximum point the function has highest value only in comparison with those values ​​that it has at all points sufficiently close to the maximum point, and at the minimum point - the smallest value only in comparison with those values ​​​​that it has at all points sufficiently close to the minimum point.

Therefore, we can clarify the above concept of extremum points of a function and call minimum points local minimum points, and maximum points local maximum points.

We look for the extrema of the function together

Example 3.

Solution: The function is defined and continuous on the entire number line. Its derivative also exists on the entire number line. Therefore, in this case, the critical points are only those at which, i.e. , from where and . Critical points and divide the entire domain of definition of the function into three intervals of monotonicity: . Let's select one control point in each of them and find the sign of the derivative at this point.

For the interval, the control point can be: find. Taking a point in the interval, we get, and taking a point in the interval, we have. So, in the intervals and , and in the interval . According to the first sufficient criterion for an extremum, there is no extremum at the point (since the derivative retains its sign in the interval), and at the point the function has a minimum (since the derivative changes sign from minus to plus when passing through this point). Let's find the corresponding values ​​of the function: , a . In the interval the function decreases, since in this interval , and in the interval it increases, since in this interval .

To clarify the construction of the graph, we find the points of intersection of it with the coordinate axes. When we obtain an equation whose roots are and , i.e., two points (0; 0) and (4; 0) of the graph of the function have been found. Using all the information received, we build a graph (see the beginning of the example).

Example 4. Find the extrema of the function and build its graph.

The domain of definition of a function is the entire number line, except for the point, i.e. .

To shorten the study, you can use the fact that this function is even, since . Therefore, its graph is symmetrical about the axis Oy and the study can only be performed for the interval.

Finding the derivative and critical points of the function:

1) ;

2) ,

but the function suffers a discontinuity at this point, so it cannot be an extremum point.

Thus, given function has two critical points: and . Taking into account the parity of the function, we will check only the point using the second sufficient criterion for an extremum. To do this, we find the second derivative and determine its sign at: we get . Since and , it is the minimum point of the function, and .

To get a more complete picture of the graph of a function, let’s find out its behavior at the boundaries of the domain of definition:

(here the symbol indicates the desire x to zero from the right, and x remains positive; similarly means aspiration x to zero from the left, and x remains negative). Thus, if , then . Next, we find

,

those. if , then .

The graph of a function has no intersection points with the axes. The picture is at the beginning of the example.

We continue to search for extrema of the function together

Example 8. Find the extrema of the function.

Solution. Let's find the domain of definition of the function. Since the inequality must be satisfied, we obtain from .

Let's find the first derivative of the function:

Let's find the critical points of the function.

(the left and right derivatives are different), however, it increases for all values ​​of x, including at points = 0.

Remark 2. Based on “softer” conditions, we can formulate a direct theorem: if the derivative of a function continuous on an interval is non-negative, then the function on this interval does not decrease. Then the direct and converse theorems in a formalized language sound like this:

function f (x) if there is a neighborhood of the point x0 such that for all x from this neighborhood f(x) ≤ f(x0).

Definition. A point x0 is called a local minimum point of the function f(x) if there is a neighborhood of the point x0 such that for all x from this neighborhood f(x) ≥ f(x0).

The value of the function at the maximum point is called the local maximum, the value of the function at the minimum point is called the local minimum of this function. The maximum and minimum of a function are called its local extrema

(extremum – extreme).

Definition. A point x0 is called a point of strict local maximum (minimum) of the function y= f(x) if for all x in the vicinity of the point x0 it is true strict inequality f(x)< f(x0 ) (соответственно

f (x) > f(x0 ) ).

Comment. In the above definition of a local extremum, we do not assume the continuity of the function at the point x 0.

maximum point, since there is a neighborhood of the point x = 0, in which f (x)< f (x 0 ).

The largest (smallest) value of a function on an interval is called global extreme. The global extremum can be reached either at the points of the local extremum or at the ends of the segment.

Necessary condition for extremum

Theorem 2. (about necessary condition extremum).

If the function y = f(x) has an extremum at the point x0, then its derivative f′ (x0) at this point is either zero or does not exist.

◄If at the point x 0 the function has an extremum and is differentiable, then at

in some neighborhood of this point, the conditions of Fermat’s theorem are satisfied, therefore, the derivative of the function at this point is equal to zero.

But the function y = f (x) may have an extremum and not be differentiable at this point. It is enough to give an example. An example would be

Note 3. Not at each of its critical points a function necessarily has a maximum or minimum.

Example 4. Consider the function y = x 3 . Critical for this function

is the point x = 0, which follows from the equationf ′ (x)= 3x 2 = 0. However, this function is increasing for all x and has no extremum.

If, when moving from the left interval to the right one, the function continues to decrease, then at point x 0 the minimum value of the function will not be reached

(no extremum).

The existence of a maximum is proved in a similar way.

In Fig. 2 a-h presents possible cases of the presence or absence of an extremum of a continuous function, the derivative of which at the critical point is equal to zero or goes to infinity.

point x = 0 changes sign infinitely many times. Therefore the function f(x) is not

is monotonically decreasing or increasing either to the left or to the right of the point x = 0.

Scheme for studying a function for an extremum:

1) find the derivative f′(x);

2) find critical points, i.e. such values x, in whichf ′ (x)= 0 or

f ′(x) = ∞;

3) examine the sign of the derivative to the left and right of each critical

changes, then at the point x 0 the function f (x) has neither a maximum nor a minimum; 4) find the values ​​of the function at extreme points.

Theorem 4. (2nd sufficient condition for extremum). Let the function y = f (x) satisfy the following conditions:

1. y = f (x) is continuous in the vicinity of the point x 0,

2. f ′ (x )= 0 at point x 0

3. f ′′ (x )≠ 0 at point x 0 .

therefore there is an extremum. The sign of the derivative changes from minus to plus, which means this is a minimum. The case f ′′ (x 0 ) is proved in a similar way< 0 .

Example 8. Examine the function y = x 2 + 2x + 3 for an extremum. Find the derivative y ′= 2x + 2 .

1) We find the critical points, for which we equate the derivative to zero: y ′= 2x + 2= 0,→ x 0 = - 1.

2) We study the sign of the derivative to the left and to the right of this point (Fig. 6).

Since the sign of the derivative changes from minus to plus, a minimum is reached at the point x = − 1.

3) Find the minimum value: ymin (− 1)= 2.

3) We examine the sign y" to the left and to the right of the point x = 0. Obviously, f ′ (x)< 0 ,

minimum of this function.

4) ymin (0)= 1.

Example 10.

Examine the function y = e -x 2 for an extremum.

1) We find the first derivative: y ′= - 2xe -x 2 .

2) Equating the derivative to zero, we find the only critical point x = 0.

3) Next we find the second derivative: y ′′= − 2e - x 2 + 4x 2 e − x 2 . Its meaning

at the point x = 0 it is equal to -2.

4) We conclude that there is a maximum of the function and calculate: y max (0)= 1.

The largest and smallest value of a function continuous on an interval

If the function f (x) is defined and continuous on the interval [a;b], then

according to Weierstrass's 2nd theorem, it reaches its maximum and minimum values ​​on this segment.

If the function f(x) takes its greatest value in internal point x 0 of the segment [a;b], then M = f (x 0) will be a local maximum of the function f (x), since in this case there is a neighborhood of the point x 0 such that the values ​​of f (x) for all points from this neighborhood will not

greater than f (x 0 ) .

However, its greatest value is M function f (x) can also take at the ends of the segment[a;b]. Therefore, in order to find the largest value M of a continuous function f (x) on the segment [a;b], we need to find all the maxima of the function in the interval (a;b) and the values ​​of f (x) at the ends of the segment [a;b] and choose

among them the largest number. Instead of limiting ourselves to finding the values Lowest value m continuous

Research to the maximum possible functions at critical points. on the segment [a;b] of the function f (x) there will be

the smallest number among all minima of the function f (x) in the interval (a;b) and values ​​f (a) and f (b).