The study of functions and their graphs is a topic that is given special attention within the framework of the high school curriculum. Some basics of mathematical analysis - differentiation - are included in the profile level of the exam in mathematics. Some schoolchildren have problems with this topic, as they confuse the graphs of the function and the derivative, and also forget the algorithms. This article will cover the main types of tasks and how to solve them.
What is the function value?
A math function is a special equation. It establishes a relationship between numbers. The function depends on the value of the argument.
The value of the function is calculated according to the given formula. To do this, substitute any argument that corresponds to the range of valid values in this formula in place of x and perform the necessary mathematical operations. What?
How can you find the smallest value of a function,using a graph function?
Graphic representation of the dependence of a function on an argument is called a function graph. It is built on a plane with a certain unit segment, where the value of a variable or argument is plotted along the horizontal abscissa axis, and the corresponding function value along the vertical ordinate axis.
The greater the value of the argument, the more to the right it lies on the graph. And the larger the value of the function itself, the higher the point is.
What does this say? The smallest value of the function will be the point that lies the lowest on the chart. In order to find it on a chart segment, you need:
1) Find and mark the ends of this segment.
2) Visually determine which point on this segment lies the lowest.
3) In response, write down its numerical value, which can be determined by projecting a point onto the y-axis.
Extremum points on the derivative chart. Where to look?
However, when solving problems, sometimes a graph is given not of a function, but of its derivative. In order to avoid accidentally making a stupid mistake, it is better to carefully read the conditions, since it depends on where you need to look for extremum points.
So, the derivative is the instantaneous rate of increase of the function. According to the geometric definition, the derivative corresponds to the slope of the tangent, which is directly drawn to the given point.
It is known that at the extremum points the tangent is parallel to the Ox axis. This means that its slope is 0.
From this we can conclude that at the extremum points the derivative lies on the x-axis or vanishes. But in addition, at these points, the function changes its direction. That is, after a period of increase, it begins to decrease, and the derivative, accordingly, changes from positive to negative. Or vice versa.
If the derivative becomes negative from positive, this is the maximum point. If from negative it becomes positive - the minimum point.
Important: if you need to specify a minimum or maximum point in the task, then in response you should write the corresponding value along the abscissa axis. But if you need to find the value of the function, then you first need to substitute the corresponding value of the argument into the function and calculate it.
How to find extremum points using derivative?
The considered examples mainly refer to the task number 7 of the exam, which involves working with a graph of a derivative or an antiderivative. But task 12 of the USE - to find the smallest value of a function on a segment (sometimes the largest) - is performed without any drawings and requires basic skills in mathematical analysis.
To perform it, you need to be able to find extremum points using the derivative. The algorithm for finding them is as follows:
- Find the derivative of a function.
- Set it to zero.
- Find the roots of the equation.
- Check if the obtained points are extremum or inflection points.
To do this, draw a diagram and onthe resulting intervals determine the signs of the derivative by substituting the numbers belonging to the segments into the derivative. If, when solving the equation, you got roots of double multiplicity, these are inflection points.
Applying the theorems, determine which points are minimum and which are maximum
Calculate the smallest value of a function using a derivative
However, having performed all these actions, we will find the values of the minimum and maximum points along the x-axis. But how to find the smallest value of a function on a segment?
What needs to be done in order to find the number that corresponds to the function at a particular point? You need to substitute the value of the argument into this formula.
Points of minimum and maximum correspond to the smallest and largest value of the function on the segment. So, to find the value of the function, you need to calculate the function using the obtained x values.
Important! If the task requires you to specify a minimum or maximum point, then in response you should write the corresponding value along the x-axis. But if you need to find the value of the function, then you must first substitute the corresponding value of the argument into the function and perform the necessary mathematical operations.
What should I do if there are no lows on this segment?
But how to find the smallest value of a function on a segment with no extremum points?
This means that the function monotonically decreases or increases on it. Then you need to substitute the value of the extreme points of this segment into the function. There are two ways.
1) Having calculatedderivative and the intervals on which it is positive or negative, to conclude whether the function is decreasing or increasing on a given segment.
In accordance with them, substitute a greater or lesser value of the argument into the function.
2) Simply substitute both points into the function and compare the resulting function values.
In which tasks finding the derivative is optional
As a rule, in the USE assignments, you still need to find the derivative. There are only a couple of exceptions.
1) Parabola.
The vertex of the parabola is found by the formula.
If a < 0, then the branches of the parabola are directed downwards. And its peak is the maximum point.
If a > 0, then the branches of the parabola are directed upwards, the vertex is the minimum point.
Having calculated the vertex point of the parabola, you should substitute its value into the function and calculate the corresponding value of the function.
2) Function y=tg x. Or y=ctg x.
These functions are monotonically increasing. Therefore, the greater the value of the argument, the greater the value of the function itself. Next, we will look at how to find the largest and smallest value of a function on a segment with examples.
Main types of tasks
Task: the largest or smallest value of the function. Example on the chart.
In the picture you see the graph of the derivative of the function f (x) on the interval [-6; 6]. At what point of the segment [-3; 3] f(x) takes the smallest value?
So, for starters, you should select the specified segment. On it, the function once takes a zero value and changes its sign - this is the extremum point. Since the derivative from negative becomes positive, it means that this is the minimum point of the function. This point corresponds to the value of the argument 2.
Answer: 2.
Continue looking at examples. Task: find the largest and smallest value of the function on the segment.
Find the smallest value of the function y=(x - 8) ex-7 on the interval [6; 8].
1. Take the derivative of a complex function.
y' (x)=(x - 8) ex-7 =(x - 8)' (ex-7) + (x - 8) (ex-7)'=1(ex-7) + (x - 8) (e x-7)=(1 + x - 8) (ex-7)=(x - 7) (ex-7 )
2. Equate the resulting derivative to zero and solve the equation.
y' (x)=0
(x - 7) (ex-7)=0
x - 7=0, or ex-7=0
x=7; ex-7 ≠ 0, no roots
3. Substitute the value of the extreme points, as well as the obtained roots of the equation, into the function.
y (6)=(6 - 8) e6-7=-2e-1
y (7)=(7 - 8) e7-7=-1e0=-11=- 1
y (8)=(8 - 8) e8-7=0e1=0
Answer: -1.
So, in this article, the main theory was considered on how to find the smallest value of a function on a segment, which is necessary for successfully solving USE tasks in specialized mathematics. Also elements of mathematicalanalysis are used when solving tasks from part C of the exam, but obviously they represent a different level of complexity, and the algorithms for their solutions are difficult to fit into the framework of one material.
