With this service you can find the greatest and smallest value functions 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
The equation f" 0 (x *) = 0 is necessary condition extremum of a function of one variable, i.e. at point x * the first derivative of the function must vanish. It highlights stationary points x s, in 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 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) . We'll find 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 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.
Finding the maximum and minimum points of a function is a fairly common task in mathematical analysis . Sometimes extremes are required. Many people think that the word “extremum” means the largest or smallest value of a function. This is not entirely true. The value may be the greatest or minimum, but not an extreme value.
Maximum happens local or global. A local maximum point is an argument that, when substituted into f(x), gives a value no less than at other points in the region around this argument. For a global maximum, this region expands to the entire region valid arguments. For the minimum, the opposite is true. An extremum is a local extreme - minimum or maximum - value.
As a rule, if mathematicians are interested in the most global great value f(x), then in the interval, not on the entire argument axis. Similar tasks usually formulated by the phrase“find the maximum point of the function on the segment.” Here it is implied that it is necessary to identify an argument in which it is no less than on the rest of the indicated segment. Search local extremum is one of the steps in solving such a problem.
Given y = f(x). It is required to determine the peak of the function on the specified segment. f(x) can reach it at the point:
- extremum, if it falls within the specified segment,
- rupture,
- limiting a given segment.
Study
The peak f(x) on a segment or interval is found by studying this function. Research plan for finding the maximum on a segment (or interval):
Now let's look at each step in detail and look at some examples.
Valid Argument Range
The region of valid arguments is those x, when substituting them into f(x) it does not cease to exist. The region of valid arguments is also called the domain of definition. For example, y = x^2 is defined on the entire argument axis. And y = 1/x is defined for all arguments except x = 0.
Finding the intersection of the region of permissible arguments and the segment (interval) under study is required in order to exclude from consideration that part of the interval where the function is not defined. For example, you need to find the minimum of y = 1/x on the interval from -2 to 2. In fact, you need to examine two half-intervals from -2 to 0 and from 0 to 2, since the equation y = 1/0 has no solution.
Asymptotes
An asymptote is a line to which the function reaches, but does not reach. If f(x) exists on the entire number line and is continuous on it, then it does not have a vertical asymptote. If it is discontinuous, then the discontinuity point is a vertical asymptote. For y = 1/x, the asymptote is given by the equation x = 0. This function reaches zero along the axis of arguments, but will reach it only by rushing into infinity.
If on the segment under study there is vertical asymptote, around which the function tends to infinity with a plus, then the peak f(x) is not determined here. And if it were determined, then the argument at which the maximum is achieved would coincide with the point of intersection of the asymptote and the argument axis.
Derivative and extrema
The derivative is function change limit when the argument changes to zero. What does it mean? Let's take a small area from the area of valid arguments and see how f(x) changes here, and then reduce this area to an infinitesimal size, in which case f(x) will change in the same way as some larger simple function, which is called the derivative.
The value of the derivative at a certain point shows at what angle the tangent to the function passes at the selected point. Negative value indicates that the function here decreases. Similarly, a positive derivative indicates an increase in f(x). This gives rise to two conditions.
1) The derivative at the extremum point is either zero or undefined. This condition is necessary, but not sufficient. Let us differentiate y = x^3, and obtain the derivative equation: y = 3*x^2. Substitute the argument “0” into the last equation, and the derivative will go to zero. However, this is not an extremum for y = x^3. It cannot have extrema; it decreases along the entire argument axis.
2) It is enough that when crossing the extremum point the derivative changes sign. That is, f(x) increases up to the maximum, and after the maximum it decreases - the derivative was positive, but became negative.
After the arguments for the local maximum have been found, they must be substituted into original equation and get maximum value f(x).
Ends of the interval and comparison of results
When searching for the maximum on a segment, you need to check the value at the ends of the segment. For example, for y = 1/x on a segment, the maximum will be at the point x = 1. Even if there is a local maximum inside the segment, there is no guarantee that the value at one of the ends of the segment will not be greater than this maximum.
Now we need to compare values at break points(if f(x) here does not tend to infinity), at the ends of the interval under study and the extremum of the function. The largest of these values will be the maximum of the function on a given section of the line.
For a problem with the wording “Find the minimum point of the function,” you need to choose the smallest of local minima and values at the ends of the interval and at break points.
Video
The function and the study of its features occupies one of the key chapters in modern mathematics. The main component of any function is graphs depicting not only its properties, but also the parameters of the derivative of this function. Let's understand this difficult topic. So what is the best way to find the maximum and minimum points of a function?
Function: definition
Any variable that in some way depends on the values of another quantity can be called a function. For example, the function f(x 2) is quadratic and determines the values for the entire set x. Let's say that x = 9, then the value of our function will be equal to 9 2 = 81.
The functions vary different types: logical, vector, logarithmic, trigonometric, numerical and others. They were studied by such outstanding minds as Lacroix, Lagrange, Leibniz and Bernoulli. Their works serve as a stronghold in modern methods studying functions. Before finding the minimum points, it is very important to understand the very meaning of the function and its derivative.
Derivative and its role
All functions depend on their variables, which means that they can change their value at any time. On the graph, this will be depicted as a curve that either falls or rises along the ordinate axis (this is the whole set of “y” numbers along the vertical graph). So, determining the maximum and minimum points of a function is precisely related to these “oscillations”. Let us explain what this relationship is.
The derivative of any function is graphed in order to study its basic characteristics and calculate how quickly the function changes (i.e. changes its value depending on the variable "x"). At the moment when the function increases, the graph of its derivative will also increase, but at any second the function can begin to decrease, and then the graph of the derivative will decrease. Those points at which the derivative changes from a minus sign to a plus sign are called minimum points. In order to know how to find minimum points, you should better understand
How to calculate derivative?
The definition and functions imply several concepts from In general, the very definition of a derivative can be expressed as follows: this is the quantity that shows the rate of change of the function.
The mathematical way of determining it seems complicated for many students, but in reality everything is much simpler. You just need to follow the standard plan for finding the derivative of any function. Below we describe how you can find the minimum point of a function without applying the rules of differentiation and without memorizing the table of derivatives.
- You can calculate the derivative of a function using a graph. To do this, you need to depict the function itself, then take one point on it (point A in the figure). Draw a line vertically down to the abscissa axis (point x 0), and at point A draw a tangent to the graph of the function. The x-axis and the tangent form a certain angle a. To calculate the value of how quickly a function increases, you need to calculate the tangent of this angle a.
- It turns out that the tangent of the angle between the tangent and the direction of the x-axis is the derivative of the function in a small area with point A. This method counts geometrically derivative definitions.
Methods for studying function
IN school curriculum In mathematics, it is possible to find the minimum point of a function in two ways. We have already discussed the first method using a graph, but how can we determine numerical value derivative? To do this, you will need to learn several formulas that describe the properties of the derivative and help convert variables type "x" into numbers. Next method is universal, so it can be applied to almost all types of functions (both geometric and logarithmic).
- It is necessary to equate the function to the derivative function, and then simplify the expression using the rules of differentiation.
- In some cases, when given a function in which the variable "x" is in the divisor, it is necessary to determine the region acceptable values, excluding the point “0” from it (for the simple reason that in mathematics you should never divide by zero).
- After this, you should transform the original form of the function into a simple equation, equating the entire expression to zero. For example, if the function looked like this: f(x) = 2x 3 +38x, then according to the rules of differentiation its derivative is equal to f"(x) = 3x 2 +1. Then we transform this expression into an equation of the following form: 3x 2 +1 = 0 .
- After solving the equation and finding the “x” points, you should plot them on the x-axis and determine whether the derivative in these sections between the marked points is positive or negative. After the designation, it will become clear at what point the function begins to decrease, that is, changes sign from minus to the opposite. It is in this way that you can find both the minimum and maximum points.
Rules of differentiation
The most basic component in studying a function and its derivative is knowledge of the rules of differentiation. Only with their help can you transform cumbersome expressions and large complex functions. Let's get acquainted with them, there are quite a lot of them, but they are all very simple due to the natural properties of both power and logarithmic functions.
- The derivative of any constant is equal to zero (f(x) = 0). That is, the derivative f(x) = x 5 + x - 160 will take the following form: f" (x) = 5x 4 +1.
- Derivative of the sum of two terms: (f+w)" = f"w + fw".
- Derivative logarithmic function: (log a d)" = d/ln a*d. This formula applies to all types of logarithms.
- Derivative of the power: (x n)"= n*x n-1. For example, (9x 2)" = 9*2x = 18x.
- The derivative of the sinusoidal function: (sin a)" = cos a. If the sin of angle a is 0.5, then its derivative is √3/2.
Extremum points
We have already discussed how to find minimum points, but there is a concept and functions. If the minimum denotes those points at which the function changes from a minus sign to a plus, then the maximum points are those points on the x-axis at which the derivative of the function changes from plus to the opposite - minus.
You can find maximum points using the method described above, but you should take into account that they indicate those areas where the function begins to decrease, that is, the derivative will be less than zero.
In mathematics, it is customary to generalize both concepts, replacing them with the phrase “points of extrema.” When a task asks you to determine these points, it means that you need to calculate the derivative of a given function and find the minimum and maximum points.
meaning
Greatest
meaning
Least
Maximum point
Minimum point
Problems of finding the points of an extremum function are solved according to a standard scheme in 3 steps.
Step 1. Find the derivative of the function
- Remember derivative formulas elementary functions and the basic rules of differentiation to find the derivative.
y′(x)=(x3−243x+19)′=3x2−243.
Step 2. Find the zeros of the derivative
- Solve the resulting equation to find the zeros of the derivative.
3x2−243=0⇔x2=81⇔x1=−9,x2=9.
Step 3. Find extreme points
- Use the interval method to determine the signs of the derivative;
- At the minimum point, the derivative is equal to zero and changes sign from minus to plus, and at the maximum point, from plus to minus.
Let's use this approach to solve the following problem:
Find the maximum point of the function y=x3−243x+19.
1) Find the derivative: y′(x)=(x3−243x+19)′=3x2−243;
2) Solve the equation y′(x)=0: 3x2−243=0⇔x2=81⇔x1=−9,x2=9;
3) The derivative is positive for x>9 and x<−9 и отрицательная при −9 How to find the largest and smallest value of a function To solve the problem of finding the largest and smallest values of a function necessary: Helps with many tasks theorem: If there is only one extremum point on a segment, and this is the minimum point, then the smallest value of the function is achieved at it. If this is a maximum point, then the greatest value is reached there. 14. Concept and basic properties of the indefinite integral. If the function f(x X, And k– number, then Briefly speaking: the constant can be taken out of the integral sign. If the functions f(x) And g(x) have antiderivatives on the interval X, That Briefly speaking: the integral of the sum is equal to the sum of the integrals. If the function f(x) has an antiderivative on the interval X, then for the interior points of this interval: Briefly speaking: the derivative of the integral is equal to the integrand. If the function f(x) is continuous on the interval X and differentiable in internal points this interval, then: Briefly speaking: the integral of the differential of a function is equal to this function plus the integration constant. Let us give a strict mathematical definition concepts of indefinite integral. An expression of the form is called integral of the function f(x)
, Where f(x)
- integrand function that is given (known), dx
- differential x
, with the symbol always present dx
. Definition. Indefinite integral called function F(x) + C
, containing an arbitrary constant C
, the differential of which is equal to integrand expression f(x)dx
, i.e. Let us remind you that - differential function and is defined as follows: Finding problem indefinite integral is to find such a function derivative which is equal to the integrand. This function is determined accurate to a constant, because the derivative of the constant is zero. For example, it is known that , then it turns out that Problem finding indefinite integral functions is not as simple and easy as it seems at first glance. In many cases, there must be skill in working with indefinite integrals, there must be experience that comes with practice and constant solving examples of indefinite integrals. It is worth considering the fact that indefinite integrals from some functions (there are quite a lot of them) are not taken in elementary functions. 15. Table of basic indefinite integrals. Basic formulas 16. Definite integral as the limit of the integral sum. Geometric and physical meaning of the integral. Let the function y=ƒ(x) be defined on the interval [a; b], a< b. Выполним следующие действия. 1. Using points x 0 = a, x 1, x 2, ..., x n = B (x 0 2. In each partial segment , i = 1,2,...,n, choose an arbitrary point with i є and calculate the value of the function in it, i.e. the value ƒ(with i). 3. Multiply the found value of the function ƒ (with i) by the length ∆x i =x i -x i-1 of the corresponding partial segment: ƒ (with i) ∆x i. 4. Let's make the sum S n of all such products: A sum of the form (35.1) is called the integral sum of the function y = ƒ(x) on the interval [a; b]. Let us denote by λ the length of the largest partial segment: λ = max ∆x i (i = 1,2,..., n). 5. Let us find the limit of the integral sum (35.1) when n → ∞ so that λ→0. If in this case the integral sum S n has a limit I, which does not depend on the method of partitioning the segment [a; b] on partial segments, nor on the choice of points in them, then the number I is called a definite integral of the function y = ƒ(x) on the segment [a; b] and is denoted Thus, The numbers a and b are called the lower and upper limits of integration, respectively, ƒ(x) - the integrand function, ƒ(x) dx - the integrand, x - the variable of integration, the segment [a; b] - area (segment) of integration. Function y=ƒ(x), for which on the interval [a; b] there is a definite integral called integrable on this interval. Let us now formulate a theorem for the existence of a definite integral. Theorem 35.1 (Cauchy). If the function y = ƒ(x) is continuous on the interval [a; b], then the definite integral Note that the continuity of a function is a sufficient condition for its integrability. However, a definite integral can also exist for some discontinuous functions, in particular for any function bounded on an interval that has a finite number of discontinuity points on it. Let us indicate some properties of the definite integral that directly follow from its definition (35.2). 1. The definite integral is independent of the designation of the integration variable: This follows from the fact that the integral sum (35.1), and therefore its limit (35.2), do not depend on what letter the argument of a given function is denoted by. 2. Definite integral with the same limits of integration equal to zero: 3. For any real number c. 17. Newton-Leibniz formula. Basic properties of a definite integral. Let the function y = f(x) continuous on the segment
And F(x) is one of the antiderivatives of the function on this segment, then Newton-Leibniz formula: The Newton-Leibniz formula is called basic formula of integral calculus. To prove the Newton-Leibniz formula, we need the concept of an integral with a variable upper limit. If the function y = f(x) continuous on the segment
, then for the argument the integral of the form is a function of the upper limit. Let's denote this function Indeed, let us write down the increment of the function corresponding to the increment of the argument and use the fifth property of the definite integral and the corollary from the tenth property: Let us rewrite this equality in the form Let's calculate F(a), using the first property of the definite integral: The increment of a function is usually denoted as To apply the Newton-Leibniz formula, it is enough for us to know one of the antiderivatives y=F(x) integrand function y=f(x) on the segment
and calculate the increment of this antiderivative on this segment. The article methods of integration discusses the main ways of finding the antiderivative. Let's give a few examples of calculating definite integrals using the Newton-Leibniz formula for clarification. Example. Calculate the value of the definite integral using the Newton-Leibniz formula. Solution. To begin with, we note that the integrand is continuous on the interval
, therefore, is integrable on it. (We talked about integrable functions in the section on functions for which there is a definite integral). From the table of indefinite integrals it is clear that for a function the set of antiderivatives for all real values of the argument (and therefore for ) is written as Now it remains to use the Newton-Leibniz formula to calculate the definite integral: 18. Geometric applications of the definite integral. GEOMETRICAL APPLICATIONS OF THE DETERMINATE INTEGRAL Calculation of body volume Calculation of the volume of a body from known areas of parallel sections: Volume of the body of rotation: ; . Example 1. Find the area of the figure bounded by the curve y=sinx by straight lines Solution: Finding the area of the figure: Example 2. Calculate the area of a figure bounded by lines Solution: Let's find the abscissa of the intersection points of the graphs of these functions. To do this, we solve the system of equations From here we find x 1 =0, x 2 =2.5. 19. The concept of differential controls. First order differential equations. Differential equation- an equation that connects the value of the derivative of a function with the function itself, the values of the independent variable, and numbers (parameters). The order of the derivatives included in the equation can be different (formally it is not limited by anything). Derivatives, functions, independent variables, and parameters may appear in an equation in various combinations, or all but one derivative may be absent altogether. Not every equation containing derivatives of an unknown function is a differential equation. For example, Partial differential equations(PDF) are equations containing unknown functions of several variables and their partial derivatives. The general form of such equations can be represented as: where are the independent variables, and is a function of these variables. The order of partial differential equations can be determined in the same way as for ordinary differential equations. Another important classification of partial differential equations is their division into equations of elliptic, parabolic and hyperbolic types, especially for second-order equations. Both ordinary differential equations and partial differential equations can be divided into linear And nonlinear. A differential equation is linear if the unknown function and its derivatives enter the equation only to the first degree (and are not multiplied with each other). For such equations, the solutions form an affine subspace of the space of functions. The theory of linear differential equations is developed much more deeply than the theory of nonlinear equations. General view of a linear differential equation n-th order: Where p i(x) are known functions of the independent variable, called coefficients of the equation. Function r(x) on the right side is called free member(the only term that does not depend on the unknown function) An important particular class of linear equations are linear differential equations with constant coefficients. A subclass of linear equations are homogeneous differential equations - equations that do not contain a free term: r(x) = 0. For homogeneous differential equations, the superposition principle holds: a linear combination of partial solutions of such an equation will also be its solution. All other linear differential equations are called heterogeneous differential equations. Nonlinear differential equations in the general case do not have developed solution methods, except for some special classes. In some cases (using certain approximations) they can be reduced to linear. For example, the linear equation of a harmonic oscillator · - homogeneous differential equation of the second order with constant coefficients. The solution is a family of functions , where and are arbitrary constants, which for a specific solution are determined from separately specified initial conditions. This equation, in particular, describes the motion of a harmonic oscillator with a cyclic frequency of 3. · Newton's second law can be written in the form of a differential equation · The Bessel differential equation is an ordinary linear homogeneous equation of the second order with variable coefficients: Its solutions are the Bessel functions. · An example of a non-homogeneous nonlinear ordinary differential equation of the 1st order: In the next group of examples there is an unknown function u depends on two variables x And t or x And y. · Homogeneous linear partial differential equation of the first order: · One-dimensional wave equation - a homogeneous linear equation in partial derivatives of the second order hyperbolic type with constant coefficients, describes the oscillation of a string if - the deflection of the string at a point with the coordinate x at a point in time t, and the parameter a sets the properties of the string: · Laplace's equation in two-dimensional space is a homogeneous linear partial differential equation of the second order of elliptic type with constant coefficients, arising in many physical problems of mechanics, thermal conductivity, electrostatics, hydraulics: · Korteweg-de Vries equation, a third-order nonlinear partial differential equation describing stationary nonlinear waves, including solitons: 20. Differential equations with separable applicable. Linear equations and Bernoulli's method. A first-order linear differential equation is an equation that is linear with respect to an unknown function and its derivative. It looks like
or 
The function is called antiderivative function. The antiderivative of a function is determined up to a constant value.
, here is an arbitrary constant.
.
, and this function is continuous and the equality is true 
.
Where .
. If we recall the definition of the derivative of a function and go to the limit at , we get . That is, this is one of the antiderivatives of the function y = f(x) on the segment
. Thus, the set of all antiderivatives F(x) can be written as 
, Where WITH– arbitrary constant.
, hence, . Let us use this result when calculating F(b): , that is 
. This equality gives the provable Newton-Leibniz formula .
. Using this notation, the Newton-Leibniz formula takes the form 
.
. Let us take the antiderivative for C=0: .
.Rectangular S.K. Function specified parametrically Polyarnaya S.K.
Calculation of areas of plane figures
Calculating the arc length of a plane curve
Calculating surface area of revolution
is not a differential equation.
can be considered as an approximation of the nonlinear equation of a mathematical pendulum 
for the case of small amplitudes, when y≈ sin y.
Where m- body weight, x- its coordinate, F(x, t) - force acting on a body with coordinate x at a point in time t. Its solution is the trajectory of the body under the action of the specified force.
