In this paper, we consider the scheme of the investigation of a function, and also give examples of research on extrema, monotonicity, and asymptotes of a given function.

- Area of existence (ODZ) function.
- Intersection of function (if any) with coordinate axes, function signs, parity, periodicity.
- Points of discontinuity (their kind). Continuity. Asymptotes are vertical.
- Monotonicity and extremum points.
- Inflection points. Convex.
- The study of the function at infinity, on the asymptotes: horizontal and oblique.
- Drawing a graph.

## Study for monotony

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

*y = 1. 3x3 \u0026 ndash; 6. 2x 2 + 5x.*

We find the intervals of constant signs*y \u0026 rsquo;*. Because the*y \u0026 rsquo;* - an elementary function, then it can change signs only at points where it turns to zero or does not exist. Her ODZ:*хєR* .

Let us find the points whose derivative is equal to 0 (zero):

## Research on extremes

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

*A*function

*g*when the value at the point is taken as the largest

*g (x*.

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

*g*on the set

*A*when the value at the lowest point

*g (x*

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

If*x _{0}* - extremum point of the function

*g*in some of its constituencies, then

*x*is called a local or local extremum point (max or min) of the function

_{0}*g.*

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

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

*Definition.* Critical are the points with a non-existent or equal to 0 (zero) derivative. It is these points that are suspicious of an extremum.

*Theorem (sufficient condition No. 1).* If the function*g* is continuous in some neighborhood of m.*x _{0}* and the sign changes through this point on passing the derivative, then the given point is an m extremum

*g*.

*Theorem (sufficient condition No. 2).* Let the function in a certain neighborhood of a point be differentiable 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 maximum (max) or minimum (min) of the function.

## Study on convexity

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

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

A function is called convex up (convex) on the interval*(a, b)*. if for any m*schech* from*(a, b)* the graph of the function on the interval lies not less than the secant line passing through the abscissas at these points.

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

If a function in a certain neighborhood of a point is continuous and*t. x _{0}* when the function changes the convexity, this point is called the inflection point of the function.

## Investigation into asymptotes

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

Asymptotes can be vertical, horizontal and inclined.

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

## Investigation of a function on a segment by the value of the smallest and largest

If the function is continuous on*[a, b]*. then by the Weierstrass theorem there exists a value of the greatest and the smallest value on this interval, that is, there exist m*glasses that belong to**[a, b]* such that*g (x _{1} ) ≤ g (x) \u0026 lt; g (x_{2} ), x_{2} [[a, b].* From theorems on monotonicity and extremum, we obtain the following scheme for investigating the function on the segment by the smallest and largest value.

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

*Comment.* If it is necessary to investigate a function on a finite interval*(a, b)*. or on an infinite*(- \u0026 gt;; b); (- ∞) + ∞)* on max and min value, then in the plan instead of the values of the function at the ends of the interval, the corresponding one-sided boundaries are searched: instead of*f (a)* are looking for*f (a +) = limf (x)*. instead*f (b)* are looking for*f (-b)*. So you can find the ODZ functions on the gap, because absolute extremes do not necessarily exist in this case.

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

- Express this value through other quantities from the condition of the problem so that it is a function of only one variable (if possible).
- Determine the interval of change of this variable.
- Perform an investigation of the function on the gap by max and min values.

*A task.* It is necessary to build a square of rectangular shape, using a meter of grid, against the wall so that on one side it adjoins the wall, and from the other three it was fenced with a grid. At what ratio of sides will the area of such a site be the largest?

*S = xy* Is a 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. 4.*

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

Let us find the other side of the rectangle:*the**= a. 2.*

Aspect ratio:*y. x = 2.*

*Answer.* The greatest 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.

## Investigation of function. Examples

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

- LDU:
*xe (- ∞) 1 U (1; ∞).* - The general form of the function (neither even nor odd), with respect to the point 0 (zero), is not symmetric.
- Signs of function. The function is elementary, therefore it can change sign only at points where it is 0 (zero), or does not exist.
- The function is elementary, therefore continuous on the DGS:
*(- ∞) 1 U (1; ∞).*

*limx 3. (1 - x) 2 = \u0026 lt;* - A break of the second kind (infinite), therefore there is a vertical asymptote at point 1;

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

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

*x = 1* - the point is critical.

*0; 3* - points are critical.

6. *y \u0026 rsquo; 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 can be inclined.

*k = 1* \u0026 ndash; number;

*b = 2* \u0026 ndash; number.

Consequently, there is an asymptote oblique*y = x + 2* at + \u0026 infin; and on -.

Given*y = (x2 + 1). (x-1). Produce and* research. Construct a graph.

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

2. *y* crosses OY (if possible) in t.*(0; g (0))*. We find*y (0) = -1* - the intersection of OY.

The intersection points of the graph with*OX* we find, solving equation*y = 0*. The equation of roots has no real roots, so this function does not intersect*OX* .

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

*g (-x) \u0026 lt; ne; g (x), and g (-x) ne; -g (x)*. This means that this is a general kind of function (neither even, nor odd).

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

5. Investigation of the function at an extremum:

and solve equation*y '= 0.*

So,*1 -? 2, 1 +? 2, 1* - critical points or points of possible extremum. These points divide the number line into four intervals*.*

At each interval, the derivative has a definite sign, which can be established by the method of intervals or the calculation of the values of the derivative at individual points. On the intervals*(- ∞ ;; 1 -? 2 ) U*

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

*1 + 2*; ∞)*xє*

*(*

*1 -? 2*; 1) U*(1;*. then the function decreases, because at these intervals the derivative is negative. Through m.

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

_{1}When passing through*x _{2}* changes the derived sign from "-" to "+", therefore, at this point there is a local minimum, and

T.*x = 1* not an extremum.

On*(- ∞ ;; 1 ) 0 > y "* . consequently, on this interval the curve is convex; if xє

*(*The curve is concave. In t

*1*; ∞)*point 1*A function is not defined, so this point is not an inflection point.

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

There are no horizontal asymptotes.

*x + 1 = y* - the asymptote is oblique by a given curve. There are no other asymptotes.

8. Taking into account the conducted studies, we construct a graph (see the figure above).