In this article, we will look at the scheme for studying a function, and also give examples of studies on extremes, monotony, and asymptotes of this function.

- Area of existence (DHS) function.
- The intersection of the function (if any) with the axes of coordinates, signs of the function, parity, periodicity.
- Break points (their genus). Continuity. Vertical asymptotes.
- Monotony and extremum points.
- Points of inflection Convex.
- The study of the function at infinity, on asymptotes: horizontal and oblique.
- Plotting.

## Monotony study

*Theorem.* If the function*g* continuous on*[a, b]*. differentiated by*(a; b)* and*g \u0026 rsquo; (x) ≥ 0 (g \u0026 rsquo; (x) ≤0)*. *xє (a; b)*. that*g* increasing (decreasing) by*[a, b]* .

*y = 1. 3x 3 \u0026 ndash; 6. 2x 2 + 5x.*

Find the intervals of permanent characters.*y \u0026 rsquo;*. Insofar as*y \u0026 rsquo;* - an elementary function, it can change signs only at points where it turns into zero or does not exist. Her DHS:*xєR* .

Find the points in which the derivative is 0 (zero):

## Extreme Study

T.*x _{0}* called the maximum point (max) on the set

*BUT*functions

*g*when the value is accepted at this point as the largest

*g (x*.

_{0}) ≥ g (x), xєAT.*x _{0}* called the minimum point (min) function

*g*on set

*BUT*when the smallest value is taken at this point as a function

*g (x*

_{0}) ≤ g (x), xєA.On set*BUT* maximum (max) and minimum (min) points are called extremum points*g*. Such extremes are also called absolute extremes on the set.

If a*x _{0}* - extremum point function

*g*in some district then

*x*referred to as a local or local extremum point (max or min) function

_{0}*g.*

*Theorem (necessary condition).* If a*x _{0}* - point of extremum (local) function

*g*. then the derivative does not exist or is equal in this t. 0 (zero).

*Definition* Critical points are points with a nonexistent or equal to 0 (zero) derivative. These data points are suspicious at the extremum.

*Theorem (condition no. 1).* If the function*g* continuous in some district t.*x _{0}* and the sign changes through this point during the transition, then this point is the extremum

*g*.

*Theorem (condition sufficient number 2).* Let the function in a district be differentiated twice and*g \u0026 rsquo; = 0, and g \u0026 rsquo; \u0026 rsquo; \u0026 gt; 0 (g \u0026 rsquo; \u0026 rsquo; \u0026 lt; 0)*. then this point is the point of maximum (max) or minimum (min) function.

## Bulge study

The function is called convex down (or concave) on the interval*(a, b)* when the graph of the function is not higher than the secant on the interval for any x with*(a, b)*. which passes through these points*.*

The function will be convex strictly down on*(a, b)*. if - the chart lies below the secant on the interval.

The function is called convex up (convex) on the gap*(a, b)*. if for any t*glasses* with*(a, b)* the graph of the function on the interval is not lower than the secant passing through the abscissas at these points.

The function will be strictly convex up on*(a, b* ), if - the graph on the interval lies above the secant.

If the function in some district of the point is continuous and through*t. x _{0}* during the transition, the function changes the convexity, then this point is called the inflection point of the function.

## Asymptote test

*Definition* Direct is called asymptote.*g (x)*. if at an infinite distance from the origin of coordinates the point of the function graph approaches it:*d (M, l).*

Asymptotes can be vertical, horizontal and oblique.

Vertical line with equation*x = x*_{0} will be the asymptote of the vertical graph of the function g. if in t. x_{0} infinite discontinuity, that is, at least one left or right boundary at this point is infinity.

## The study of the function on the segment on the value of the smallest and largest

If the function is continuous on*[a, b]*. then by the Weierstrass theorem there is the largest value and the smallest value on this segment, that is, there are t*points that belong**[a, b]* such that*g (x _{1} ) ≤ g (x) \u0026 lt; g (x_{2} ), x_{2} є [a, b].* From the theorems on monotonicity and extrema, we obtain the following scheme for studying a function on a segment for the smallest and largest value.

- Find derivative
*g \u0026 rsquo; (x)*. - Search function value
*g*at these points and at the ends of the segment. - The values found compare and select the smallest and largest.

*Comment.* If you want to make a study of the function on a finite interval*(a, b)*. or at infinite*(- \u0026 infin ;; b); (- \u0026 infin ;; + \u0026 infin;)* on max and min value, then in the plan, instead of the values of the function at the ends of the gap, they search for the corresponding one-sided boundaries: instead of*f (a)* are looking for*f (a +) = limf (x)*. instead*f (b)* are looking for*f (-b)*. So you can find the LDU function on the interval, because absolute extremes do not necessarily exist in this case.

## Application of the derivative to the solution of applied problems on the extremum of certain quantities

- Express this value in terms of other values from the condition of the problem so that it is a function of only one variable (if possible).
- Determine the period of change of this variable.
- Conduct a study of the function on the interval at max and min values.

*Task.* It is necessary to build a rectangular platform, using grid meters, against the wall so that on one side it fits to the wall, and on the other three it is fenced with a grid. At what aspect ratio will the area of such a site be greatest?

*S = xy* - function of 2 variables.

*S = x (a - 2x)* - function of the 1st variable*; x є [0; a: 2].*

*S = ax - 2x 2; S ’= a - 4x = 0, xєR, x = a. four.*

*S (a. 4) = a 2. 8* - the greatest value;

Find the other side of the rectangle:*at**= a. 2*

Aspect ratio:*y. x = 2.*

*Answer.* The largest area will be equal to*a 2/8*. if the side that is parallel to the wall is 2 times larger than the other side.

## Research function. Examples

There is*y = x 3. (1-x) 2. Perform research.*

- DHS:
*xє (- inf; 1) U (1; inf);* - The general form of the function (neither even nor odd), is not symmetric about the point 0 (zero).
- Signs of function. The function is elementary, so it can only change sign at the points where it is 0 (zero), or does not exist.
- The function is elementary, therefore continuous on the DHS:
*(- \u0026 infin ;; 1) U (1; \u0026 infin;).*

*limx 3. (1- x) 2 = \u0026 infin;* - Gap of the 2nd kind (infinite), therefore there is a vertical asymptote at point 1;

*x = 1* - the equation of the vertical asymptote.

5. *y \u0026 rsquo; = x 2 (3 - x). (1 - x) 3;*

*x = 1* - The point is critical.

*0; 3* - critical points.

6. *y \u0026 rsquo; \u0026 rsquo; = 6x. (1 - x) 4;*

Critical t.*1, 0;*

7. *limx 3. (1 - 2x + x 2) = \u0026 infin;* - there is no horizontal asymptote, but it may be oblique.

*k = 1* \u0026 ndash; number;

*b = 2* \u0026 ndash; number.

Therefore, there is an asymptote inclined*y = x + 2* by + \u0026 infin; and on - \u0026 infin ;.

Given*y = (x 2 + 1). (x - 1). Produce and* investigation Build a graph.

1. The domain of existence is the whole numerical line, except for m.*x = 1**.*

2. *y* Crosses OY (if possible) in t.*(0; g (0))*. Find*y (0) = -1* - t. Intersection OY.

Points of intersection of the graph with*OX* find by solving the equation*y = 0*. The root equation has no real, therefore this function does not intersect*OX* .

3. The function is non-periodic. Consider the expression

*g (-x) ne g (x) and g (-x)? ne -g (x)*. This means that this is a generic function (neither even nor odd).

4. T.*x = 1* the gap has a second kind. At all other points, the function is continuous.

5. The study of the function on the extremum:

and solve the equation*y ’= 0.*

So,*1 - \u0026 radic; 2, 1 + \u0026 radic; 2, 1* - Critical points or points of possible extremum. These points break the number line into four intervals.*.*

At each interval, the derivative has a certain sign, which can be set by the method of intervals or by calculating the values of the derivative at individual points. At intervals*(- \u0026 infin ;; 1 - \u0026 radic; 2 ) U*

*(*. positive derivative, so the function grows; if a

*1 + rad 2*; \u0026 infin;)*xє*

*(*

*1 - \u0026 radic; 2*; 1) U*(1;*. then the function decreases, because at these intervals the derivative is negative. Through t.

*1 + rad 2*)*x*during the transition (the movement follows from left to right) changes the derivative sign from “+” to “-“, therefore, at this point there is a local maximum, we find

_{1}When going through*x _{2}* changes the derivative sign from “-” to “+”, therefore, at this point there is a local minimum, and

T.*x = 1* not t extremum.

On*(- \u0026 infin ;; 1 ) 0 > y ”* . consequently, on this interval the curve is convex; if xє

*(*- The curve is concave. In t

*1*; \u0026 infin;)*1 point*no function is defined, so this point is not an inflection point.

7. From the results of paragraph 4 it follows that*x = 1* - asymptote vertical curve.

Horizontal asymptotes are absent.

*x + 1 = y* - the asymptote is inclined by this curve. There are no other asymptotes.

8. Considering the conducted studies, we build a graph (see the figure above).