(Fig.1)
Figure 1. Schedule derivative
Properties of graphics derivative
- At intervals of increasing, the derivative is positive. If the derivative at a certain point from a certain interval has a positive value, then the function graph at this interval increases.
- At the flattening intervals, the derivative is negative (with a minus sign). If the derivative at a certain point from some interval has a negative value, then the function schedule decreases at this interval.
- The derivative at the point x is equal to the angular coefficient of tangential, conducted to the graph of the function at the same point.
- At the maximum minimum points, the derivative function is zero. Tanner to the graphics of the function at this point parallel to the axis oh.
Example 1.
According to the graph (Fig. 2), the derivative is determined at which point on the segment [-3; 5] The function is maximum.
Figure 2. Derivative graph
Solution: On this segment, the derivative is negative, and therefore the function decreases from left to right, and is the largest value from the left side at point -3.
Example 2.
According to graphics (Fig. 3), the derivative determines the number of maximum points on the segment [-11; 3].
Figure 3. Derivative graph
Solution: The maximum points correspond to the point of change of the sign of the derivative with a positive to negative. At this interval, the function changes twice the sign from the plus on the minus - at point 10 and at point -1. So the number of maximum points is two.
Example 3.
According to graphics (Fig. 3), the derivative determines the number of points of the minimum of the segment [-11; -one].
Solution: Minimum points correspond to the point shift points with a negative to positive. On this segment such a point is only -7. So, the number of points on the minimum on a given segment is one.
Example 4.
According to graphics (Fig. 3), the derivative determines the number of extremum points.
Solution: the extremum is the points of both the minimum and the maximum. We find the number of points in which the derivative changes the sign.
The figure shows a graph of the derivative function f (x) defined in the interval [-5; 6]. Find the number of points of the graph F (X), in each of which the tangent, spent on the graph of the function, coincides or parallel to the abscissa axis
The figure shows a graph of the derivative of the differentiable function Y \u003d F (x).
Find the number of points of the function of the function belonging to the segment [-7; 7], in which the function tangent of the function is parallel to the direct specified equation y \u003d -3x.
The material point M begins to move from point A and moving in a straight line for 12 seconds. The schedule shows how the distance from the point A to the point M over time. At the abscissa axis, the time T is postponed in seconds, on the ordinate axis - distance s in meters. Determine how many times during the movement the speed point M appealed to zero (the beginning and end of the movement do not take into account).
The figure shows sections of the function of the function y \u003d f (x) and tangent to it at a point with an abscissa x \u003d 0. It is known that this tannative parallel to the direct passing through the points of the graph with the abscissions x \u003d -2 and x \u003d 3. Using it, Find the value of the derivative F "(O).
The figure shows a graph Y \u003d F '(X) - the derivative of the function f (x) defined on the segment (-11; 2). Find the abscissue of the point in which the tangent of the function of the function y \u003d f (x) is parallel to the abscissa axis or coincides with it.
The material point moves straightly according to the law x (t) \u003d (1/3) t ^ 3-3t ^ 2-5t + 3, where X is the distance from the point of reference in meters, T is the time in seconds measured from the beginning of the movement. At what point of time (in seconds), its speed was 2 m / s?
The material point moves along a straight line from the initial to the end position. The figure shows the schedule of its movement. At the abscissa axis, the time is postponed in seconds, on the axis of the ordinate - the distance from the initial position of the point (in meters). Find the average speed of the point. Give the answer in meters per second.
The function y \u003d f (x) is determined on the interval [-4; four]. The figure shows the graph of its derivative. Find the number of points of the function of the function y \u003d f (x), the tangent of which forms an angle of 45 ° with the positive direction of the axis.
The function y \u003d f (x) is defined on the segment [-2; four]. The figure shows the schedule for its derivative. Find the abscissa of the graph of the function of the function y \u003d f (x) in which it takes the smallest value on the segment [-2; -0,001].
The figure shows the graph of the function y \u003d f (x) and tangent to this graphics spent at the point x0. The tangent is set by the Y \u003d -2X + 15 equation. Find the value of the derivative function y \u003d - (1/4) f (x) + 5 at point x0.
On the graph of the differential function y \u003d f (x) seven dots are marked: x1, .., x7. Find all marked points in which the derivative function f (x) is greater than zero. In response, specify the number of these points.
The figure shows a graph y \u003d f "(x) derivative function F (x), determined on the interval (-10; 2). Find the number of points in which the function of the function f (x) is parallel to the direct Y \u003d -2x-11 or coincides with it.
The figure shows the graph Y \u003d F "(x) - the derivative function f (x). On the abscissa axis, nine points are noted: x1, x2, x3, x4, x5, x6, x9, x7, x8, x9.
How many of these points belongs to the decreases of the function F (X)?
The figure shows the graph of the function y \u003d f (x) and tangent to this graphics spent at point x0. The tangent is given by the equation y \u003d 1.5x + 3.5. Find the value of the derivative function y \u003d 2f (x) - 1 at point x0.
The figure shows a graph Y \u003d F (x) of one of the primitive functions F (X). The graph marked six points with the abscissions x1, x2, ..., x6. Whole of these points, the function y \u003d f (x) takes negative values?
The figure shows the car motion schedule along the route. At the abscissa axis, time is postponed (in hours), on the axis of the ordinate - the path passed (in kilometers). Find the average vehicle speed on this route. Answer in km / h
The material point moves straightly according to the law x (t) \u003d (- 1/6) t ^ 3 + 7t ^ 2 + 6t + 1, where X is the distance from the point of reference (in meters), T is the movement time (in seconds). Find her speed (in meters per second) at time t \u003d 6 s
The figure shows a graph of the primitive y \u003d f (x) of some function y \u003d f (x), determined on the interval (-6; 7). Using the pattern, determine the number of zeros of the function f (x) at this interval.
The figure shows a graph y \u003d f (x) one of the primary some functions F (x) defined on the interval (-7; 5). Using the pattern, determine the amount of solutions of the equation F (x) \u003d 0 on the segment [- 5; 2].
The figure shows a graph of the differential function y \u003d f (x). On the abscissa axis, nine points are marked: x1, x2, ... x9. Find all the marked points in which the derivative function f (x) is negative. In response, specify the number of these points.
The material point moves straightforwardly by law x (t) \u003d 12t ^ 3-3t ^ 2 + 2t, where x is the distance from the point of reference in meters, T is the time in seconds measured from the beginning of the movement. Find her speed (in meters per second) at time t \u003d 6 s.
The figure shows the graph of the function y \u003d F (x) and tangent to this graphics spent at the point x0. The tangent equation is shown in the figure. Find the value of the derivative function y \u003d 4 * f (x) -3 at point x0.
Imagine a straight road passing through a hilly area. That is, it goes up, then down, but right or left does not turn. If the axis is directed along the road horizontally, and - vertically, then the line of the road will be very similar to a schedule of some continuous function:
The axis is a certain level of zero height, we use the level of the sea as it.
Moving forward on such a road, we also move up or down. We can also say: when the argument is changed (advanced along the abscissa axis) the value of the function changes (movement along the ordinate axis). And now let's think about how to determine the "steepness" of our road? What could it be for the magnitude? Very simple: how much will the height change when moving forward for a certain distance. After all, at different parts of the road, moving forward (along the abscissa axis) for one kilometer, we will rise or fall on a different number of meters relative to the sea level (along the ordinate axis).
Promotion forward to be denoted (read "Delta X").
The Greek letter (Delta) in mathematics is usually used as a prefix meaning "change". That is - this is a change in the value - change; Then what is? That's right, change value.
Important: Expression is a single integer, one variable. You can never tear off the "Delta" from "IKSA" or any other letter! That is, for example.
So, we advanced forward, horizontally, on. If the line of the road we compare the function with a graph, then how do we designate the rise? Sure, . That is, when moving forward on we rising above.
It is easy to calculate the amount: if at the beginning we were at the height, and after moving were at the height, then. If the end point turned out to be lower than the initial, it will be negative - this means that we do not go up, but let down.
Let's go back to the "steepness": this is the value that shows how much strongly (cool) increases the height when moving forward per unit distance:
Suppose that on a site of the path when moving at km the road rises upwards at km. Then the steepness in this place is equal. And if the road when promoting on m sank to km? Then the steep is equal to.
Now consider the top of some hill. If you take the beginning of the site for half a kilometer to the top, and the end - after half a kilometer after it, it can be seen that the height is almost the same.
That is, in our logic it turns out that the steepness here is almost equal to zero, which is clearly not true. Just at a distance in km can change a lot. It is necessary to consider smaller sections for a more adequate and accurate assessment of the steepness. For example, if you measure the change in the height when moving to one meter, the result will be much more accurate. But this accuracy may not be enough for us - because if there is a pillar in the middle of the road, we can simply slip it. What distance then choose? Centimeter? Millimeter? Less is better!
In real life, measuring the distance with an accuracy to the miliethra - more than enough. But mathematicians always strive for excellence. Therefore, the concept was invented infinitely smallThat is, the magnitude of the module is less than any number that can only be called. For example, you say: one trillion! Where is less? And you filed this number on - and it will be even less. Etc. If we want to write that the magnitude is infinitely small, we write like this: (I read "X is striving for zero"). It is very important to understand that this number is not zero! But very close to it. This means that it can be divided into it.
The concept opposite is infinitely small - infinitely large (). You already probably snapped with him when I was engaged in inequalities: this is the number of module more than any number that can be invented. If you came up with the largest of the possible numbers, just multiply it to two, and it will turn out even more. And infinity even more than what happens. In fact, the infinitely large and infinitely small reversed each other, that is, when, and on the contrary: when.
Now back to our road. The perfectly counted steepness is a bengeon, calculated for an infinitely small segment of the path, that is:
I note that with an infinitely small movement, the change in the height will also be infinitely small. But I remind you, infinitely small - does not mean equal to zero. If you share infinitely small numbers each other, it may be quite a common number, for example. That is, one low value can be exactly more than once more.
What is all this? The road, steepness ... We are not going to go to the rally, and we learn mathematics. And in mathematics everything is just the same, only called differently.
The concept of derivative
The derivative of the function is the ratio of the increment of the function to the increment of the argument with an infinitely small increment of the argument.
Increment In mathematics call change. How much the argument changed () when moving along the axis is called increment of argument and referred to how much the function changed (height) when moving forward along the axis is called, called increment of function and is denoted.
So, the derived function is attitude to when. We indicate the derivative of the same letter as the function, only with the stroke on the right: or simply. So, we will write the derivative formula using these notation:
As in analogy with expensive here, with an increase in the function, the derivative is positive, and when decreasing is negative.
Does the derivative happen to zero? Sure. For example, if we are going along a flat horizontal road, the steep is zero. And the truth is, the height is not entirely changing. So with the derivative: the derivative of the constant function (constant) is zero:
since the increment of such a function is zero at any.
Let's remember the example from the hill. It turned out that it was possible that you can position the ends of the segment along different directions from the vertex that the height at the ends turns out to be the same, that is, the segment is located in parallel axis:
But large segments are a sign of inaccurate measurement. We will raise our cut up parallel to yourself, then its length will decrease.
In the end, when we are infinitely close to the top, the length of the segment will become infinitely small. But at the same time, it remained parallel to the axis, that is, the height difference at its ends is zero (does not seek, namely equal to). So derivative
It is possible to understand this: when we stand on the top of the top, the little displacement to the left or right changes our height is negligible.
There is a purely algebraic explanation: the left of the top is the function increases, and to the right - decreases. As we have already found out earlier, with an increase in the function, the derivative is positive, and as descending, is negative. But it changes smoothly, without jumps (because the road does not change the slope anywhere). Therefore, between negative and positive values \u200b\u200bmust be. He will be where the function neither increases, nor decreases - at the point of the vertex.
The same is true for the depression (the area where the function on the left decreases, and on the right - increases):
A little more about increments.
So, we change the argument by magnitude. Change from what value? What is he (argument) now? We can choose any point, and now we will dance from it.
Consider a point with the coordinate. The value of the function in it is equal. Then make something increment: increase the coordinate on. What is the argument now? Very easy: . And what is the value of the function now? Where the argument, there and the function :. And what about the increment of the function? Nothing new: it is still the magnitude of which the function has changed:
Practice to find increments:
- Find the increment of the function at the point when the argument is increasing.
- The same for the function at the point.
Solutions:
At different points at one and the same increment of the argument, the increment of the function will be different. It means that the derivative at every point is its own (we discussed at the very beginning - the steepness of the road at different points is different). Therefore, when we write a derivative, you must specify at what point:
Power function.
The power is called the function where the argument is to some extent (logical, yes?).
Moreover, to either :.
The simplest case is when the degree indicator:
We find its derivative at the point. We remember the definition of the derivative:
So, the argument changes from before. What is the increment of the function?
Increment is. But the function at any point is equal to its argument. Therefore:
The derivative is equal to:
Derived from equal:
b) Now consider the quadratic function () :.
And now remember that. This means that the value of increment can be neglected, since it is infinitely small, and therefore insignificantly against the background of another term:
So, we were born next rule:
c) We continue the logical range :.
This expression can be simplified in different ways: to reveal the first bracket by the formula of the abbreviated multiplication of the cube amount, or decompose the entire expression on the factors by the Cube difference formula. Try to do it yourself by any of the proposed ways.
So, I got the following:
And again remember that. This means that you can neglect by all the terms containing:
We get :.
d) similar rules can be obtained for large degrees:
e) It turns out that this rule can be generalized for a power function with an arbitrary indicator, not even:
| (2) |
You can formulate the rule with words: "The degree is taken forward as a coefficient, and then decreases by".
Let us prove this rule later (almost at the very end). And now consider a few examples. Find derived functions:
- (in two ways: by the formula and using the derivative determination - considering the increment of the function);
Trigonometric functions.
Here we will use one fact of the highest mathematics:
When expressing.
Proof You will know in the first year of the Institute (and to be there, you need to pass it well). Now just show it graphically:
We see that when the function does not exist - the point on the graph of the population. But the closer to the value, the closer the function to. This is the most "striving."
You can additionally check this rule using the calculator. Yes, yes, do not be shy, take a calculator, we are not on the exam yet.
So try :;
Do not forget to transfer the calculator to "Radian" mode!
etc. We see that the smaller, the closer the value of the relationship to.
a) Consider the function. As usual, we will find its increment:
Turn the difference in sines into the work. To do this, we use the formula (remember the topic "") :.
Now the derivative:
We will replace :. Then, with infinitely small, it is also infinitely small :. The expression for takes the form:
And now you remember that when expressing. And also that if the infinitely low value can be neglected in the amount (that is, when).
So, we get the following rule: sinus derivative equal to cosine:
This is basic ("tabular") derivatives. Here they are one list:
Later we add to them a few more, but these are the most important, as they are most often used.
Practice:
- Find derived function at point;
- Find derived function.
Solutions:
Exhibitor and natural logarithm.
There is such a function in mathematics, the derivative of which with any equal value of the function itself at the same way. It is called "Exhibitor", and is an indicative function
The basis of this function is a constant is an infinite decimal fraction, that is, the number is irrational (such as). It is called "the number of Euler", therefore and denote the letter.
So, the rule:
Remember very easy.
Well, let's not go far, we will immediately consider the reverse function. What function is the reverse for an indicative function? Logarithm:
In our case, the basis is the number:
Such a logarithm (that is, a logarithm with a base) is called "natural", and for it we use a special designation: instead of writing.
What is equal to? Of course, .
The derivative of the natural logarithm is also very simple:
Examples:
- Find derived function.
- What is the derived function equal?
Answers: Exhibitor and natural logarithm - the functions are uniquely simple from the point of view of the derivative. Exchange and logarithmic functions with any other base will have another derivative, which we will analyze later with you, after passing the differentiation rules.
Differentiation rules
Rules What? Again the new term, again?! ...
Differentiation - This is the process of finding a derivative.
Only and everything. And how else to name this process in one word? Not a production of ... The differential of mathematics is called the most increment of the function at. This term is happening from Latin Differentia - a difference. Here.
When displaying all these rules, we will use two functions, for example, and. We will also need formulas for their increments:
Total there are 5 rules.
The constant is made out of the sign of the derivative.
If - some kind of constant number (constant), then.
Obviously, this rule works for difference :.
We prove. Let, or easier.
Examples.
Find derived functions:
- at the point;
- at the point;
- at the point;
- at point.
Solutions:
Derived work
Here everything is similar: we introduce a new function and find its increment:
Derivative:
Examples:
- Find derivatives of functions and;
- Find the function derivative at the point.
Solutions:
Derivative indicative function
Now your knowledge is enough to learn how to find a derivative of any indicative function, and not just exhibitors (not forgotten what it is?).
So, where is some number.
We already know the derivative function, so let's try to bring our function to a new base:
To do this, we use a simple rule :. Then:
Well, it turned out. Now try to find a derivative, and do not forget that this feature is complex.
Happened?
Here, check yourself:
The formula turned out to be very similar to the derivative exhibit: as it was, it remained, only a multiplier appeared, which is just a number, but not a variable.
Examples:
Find derived functions:
Answers:
Derivative logarithmic function
Here is similar: you already know the derivative from the natural logarithm:
Therefore, to find an arbitrary from logarithm with another reason, for example:
You need to bring this logarithm to the base. And how to change the basis of the logarithm? I hope you remember this formula:
Only now instead we will write:
In the denominator, it turned out just a constant (constant number, without a variable). The derivative is very simple:
The derivatives of the indicative and logarithmic functions are almost not found in the exam, but it will not be superfluous to know them.
Derivative complex function.
What is a "complex function"? No, it is not a logarithm, and not Arcthangence. These functions can be complex for understanding (although if the logarithm seems to you difficult, read the topic "Logarithms" and everything will pass), but from the point of view of mathematics the word "complex" does not mean "difficult".
Imagine a small conveyor: two people are sitting and have some kind of actions with some objects. For example, the first wraps a chocolate in the wrapper, and the second implies it with a ribbon. It turns out such an integral object: a chocolate, wrapped and lined with a ribbon. To eat a chocolate, you need to do reverse action in reverse order.
Let's create a similar mathematical conveyor: first we will find a cosine of the number, and then the resulting number to be erected into a square. So, we give a number (chocolate), I find his cosine (wrap), and then you will be erected by what I did, in a square (tie to the ribbon). What happened? Function. This is an example of a complex function: when to find its meanings we do the first action directly with the variable, and then another action with what happened as a result of the first one.
We can completely do the same actions and in the reverse order: first you will be erected into a square, and then I'm looking for a cosine of the resulting number :. It is easy to guess that the result will be almost always different. An important feature of complex functions: when the procedure change, the function changes.
In other words, a complex function is a function, the argument of which is another feature.: .
For the first example,.
The second example: (the same). .
Action that we do the latter will call "External" function, and the action performed first - respectively "Internal" function (These are informal names, I use them only to explain the material in simple language).
Try to determine myself what function is external, and which is internal:
Answers:The separation of internal and external functions is very similar to replacement of variables: for example, in function
we produce a replacement of variables and get a function.
Well, now we will extract our chocolate chocolate - search for a derivative. The procedure is always reverse: first we are looking for an external function derivative, then multiply the result on the derivative of the internal function. With regard to the original example, it looks like this:
Another example:
So, we finally formulate the official rule:
The algorithm for finding a derivative complex function:
It seems to be simple, yes?
Check on the examples:
DERIVATIVE. Briefly about the main thing
Derived function - The ratio of the increment of the function to the increment of the argument with an infinitely small increment of the argument:
Basic derivatives:
Differentiation Rules:
The constant is made for the sign of the derivative:
Derived amount:
Production work:
Private derivative:
Derivative complex function:
Algorithm for finding a derivative of complex function:
- We define the "internal" function, we find its derivative.
- We define the "external" function, we find its derivative.
- Multiply the results of the first and second items.
Well, the topic is finished. If you read these lines, then you are very cool.
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(Fig.1)
Figure 1. Schedule derivative
Properties of graphics derivative
- At intervals of increasing, the derivative is positive. If the derivative at a certain point from a certain interval has a positive value, then the function graph at this interval increases.
- At the flattening intervals, the derivative is negative (with a minus sign). If the derivative at a certain point from some interval has a negative value, then the function schedule decreases at this interval.
- The derivative at the point x is equal to the angular coefficient of tangential, conducted to the graph of the function at the same point.
- At the maximum minimum points, the derivative function is zero. Tanner to the graphics of the function at this point parallel to the axis oh.
Example 1.
According to the graph (Fig. 2), the derivative is determined at which point on the segment [-3; 5] The function is maximum.
Figure 2. Derivative graph
Solution: On this segment, the derivative is negative, and therefore the function decreases from left to right, and is the largest value from the left side at point -3.
Example 2.
According to graphics (Fig. 3), the derivative determines the number of maximum points on the segment [-11; 3].
Figure 3. Derivative graph
Solution: The maximum points correspond to the point of change of the sign of the derivative with a positive to negative. At this interval, the function changes twice the sign from the plus on the minus - at point 10 and at point -1. So the number of maximum points is two.
Example 3.
According to graphics (Fig. 3), the derivative determines the number of points of the minimum of the segment [-11; -one].
Solution: Minimum points correspond to the point shift points with a negative to positive. On this segment such a point is only -7. So, the number of points on the minimum on a given segment is one.
Example 4.
According to graphics (Fig. 3), the derivative determines the number of extremum points.
Solution: the extremum is the points of both the minimum and the maximum. We find the number of points in which the derivative changes the sign.
The derivative function is one of the complex themes in the school program. Not every graduate will answer the question of what is derived.
This article is simply clearly talking about what a derivative is and for what it needs. We will not strive to strive for the mathematical strictness of the presentation. The most important thing is to understand the meaning.
We remember the definition:
The derivative is the speed of change of function.
In the picture - graphics of three functions. What do you think is growing faster?
The answer is obvious - the third. It has the greatest speed of change, that is, the greatest derivative.
Here is another example.
Kostya, Grisha and Matvey simultaneously got a job. Let's see how their income changed during the year:
On the schedule immediately everything can be seen, is not it? The bone's income for half a year has grown more than twice. And Grisha revenue has also grown, but quite a little bit. And Matthew's income decreased to zero. Starting conditions are the same, and the speed of change of function, that is derivative- Different. As for Matthew - his income is negatively derived.
Intuitively, we are easily assessing the speed of change of function. But how do you do it?
In fact, we look at how cool the graph of the function goes up (or down). In other words, how quickly changes y with a change in x. Obviously, the same function at different points can have a different value of the derivative - that is, it may vary faster or slower.
The derivative function is indicated.
Show how to find using the graph.
A graph is drawn some function. Take a point with an abscissa on it. We draw at this point tangent to the graphics function. We want to evaluate how cool up a graph of a function. Comfortable value for this - tangent tilt angle.
The derivative of the function at the point is equal to the tangent of the tilt angle, carried out to the graph of the function at this point.
Please note - as an angle of tagging tangent, we take an angle between the tangent and positive direction of the axis.
Sometimes students ask what tangent to the function graphics. This is a direct, having a single common point with a schedule on this plot, and as shown in our figure. Looks like a tangent to the circumference.
We will find. We remember that the tangent of an acute angle in a rectangular triangle is equal to the attitude of the opposite catech to the adjacent one. From the triangle:
We found a derivative with the help of a graph, not even knowing the formula function. Such tasks are often found in the exam in mathematics at the number.
There is another important ratio. Recall that the direct is given by the equation
The value in this equation is called angular coefficient direct. It is equal to the tangent of the angle of inclination direct to the axis.
.
We get that
We remember this formula. It expresses the geometric meaning of the derivative.
The derivative of the function at the point is equal to the angular coefficient of tangent, carried out to the graph of the function at this point.
In other words, the derivative is equal to tangent tilt angle.
We have already said that the same function at different points may have a different derivative. Let's see how the derivative is associated with the behavior of the function.
Draw a graph of some function. Let this function be increasing on some sections, on others - decreases, with different speeds. And even if this feature there will be a point of maximum and a minimum.
At the point, the function increases. Tangent to the graph, spent at the point forms a sharp angle; With a positive axis direction. So, at the point the derivative is positive.
At the point, our function decreases. Tangent at this point forms a stupid angle; With a positive axis direction. Since the dull angle tangent is negative, a derivative is negative at the point.
That's what it turns out:
If the function increases, its derivative is positive.
If decreases, its derivative is negative.
And what will be at the points of the maximum and minimum? We see that at points (maximum point) and (minimum point) tangent horizontal. Consequently, the tangent tangent tilt angle at these points is zero, and the derivative is also zero.
Point is a maximum point. At this point, the increasing function is replaced by descending. Consequently, the sign of the derivative changes at a point with a "plus" to "minus".
At the point - the point of the minimum - the derivative is also zero, but its sign changes from "minus" to the "plus".
Conclusion: With the help of a derivative, you can learn about the behavior of the function all that interests us.
If the derivative is positive, then the function increases.
If the derivative is negative, the function decreases.
At the point of the maximum, the derivative is zero and changes the sign from the "plus" to "minus".
At the point of the minimum, the derivative is also zero and changes the sign from "minus" to the "plus".
We write these conclusions in the form of a table:
| increases | maximum point | decrease | point of minimum | increases | |
| + | 0 | - | 0 | + |
We will make two small clarifications. One of them will be needed to you when solving the problem. Other - in the first year, with a more serious study of functions and derivatives.
A case is possible when the derivative of the function at some point is zero, but no maximum, no minimum function at this point at this point. This is the so-called :
At the point tangent to the graphics of the horizontal, and the derivative is zero. However, the function of the function increased - and after the point continues to increase. The sign of the derivative does not change - it has been positive and remained.
It also happens that at the point of the maximum or minimum, the derivative does not exist. On the chart, it corresponds to a sharp breaking when the tangent is impossible at this point.
And how to find a derivative if the function is not specified by the schedule, but by the formula? In this case, applied
