The study of functions and their schedules is a topic that receives particular attention in the framework of the high school curriculum. Some foundations of mathematical analysis - differentiation - are included in the profile level of the exam in mathematics. Some students have problems with this topic, as they confuse the graphs of the function and the derivative, and also forget the algorithms. This article will discuss the main types of tasks and how to solve them.
What is the value of a function?
Mathematical function is a special equation. It establishes the 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 permissible values in this formula in place of x and perform the necessary mathematical operations. What kind?
How can I find the smallest function value using the function graph?
A graphical representation of the dependence of a function on an argument is called a function graph. It is built on a plane with a specific unit segment, where the horizontal axis of the abscissa represents the value of the variable or argument, and the vertical axis of the ordinate represents the corresponding value of the function.
The larger the value of the argument, the more to the right it lies on the chart. And the greater the value of the function itself, the higher the point.
What is this talking about? The smallest value of the function will be the point that lies below everything on the graph. In order to find it on the graph segment, you need:
1) Find and mark the ends of this segment.
2) Visually determine which point on this segment is the lowest.
3) In response, write down its numerical value, which can be determined by projecting a point on the ordinate axis.
Extremum points on the derivative graph. Where to looking for?
However, when solving problems, sometimes a graph is given not of a function, but of its derivative. In order to accidentally prevent a stupid mistake, it is better to carefully read the conditions, since it depends on where you need to look for extreme points.
So, the derivative is the instantaneous rate of increase of the function. According to the geometric definition, the derivative corresponds to the angular coefficient of the tangent, which is directly drawn to this point.
It is known that at 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 points of the extremum the derivative lies on the abscissa axis or vanishes. But in addition, at these points the function changes its direction. That is, after a period of increase begins to decrease, and the derivative, respectively, is replaced from positive to negative. Or vice versa.
If the derivative of positive becomes negative, this is the maximum point. If the negative becomes positive - the minimum point.
Important: if the task requires you to specify a minimum or maximum point, then the corresponding value along the abscissa should be recorded in response. But in case you need to find the value of the function, then first you need to substitute the corresponding value of the argument into the function and calculate it.
How to find extremum points using a derivative?
The examples considered mainly relate to task number 7 of the exam, which involves working with a schedule of a derivative or antiderivative. But task 12 of the Unified State Examination - 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 do this, you need to be able to find extremum points using the derivative. The algorithm for finding them is as follows:
- Find the derivative of the function.
- Set it to zero.
- Find the roots of the equation.
- Check if the resulting points are points of extremum or inflection.
To do this, draw a diagram and determine the signs of the derivative at the resulting intervals, substituting the numbers belonging to the segments in the derivative. If, when solving the equation, you get roots of double multiplicity, these are the inflection points.
- Applying the theorems, determine which points are the minimum points and which are the maximum points.
Calculating the smallest value of a function using a derivative
However, having completed all these actions, we will find the values of the minimum and maximum points along the abscissa. But how to find the smallest function value on a segment?
What needs to be done in order to find the number that the function corresponds to at a particular point? It is necessary to substitute the value of the argument in this formula.
The minimum and maximum points 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 the corresponding value along the abscissa should be recorded in response. But in case 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 to do if there are no minimum points on this segment?
But how to find the smallest value of a function on a segment on which there are no extremum points?
This means that on it the function monotonically decreases or increases. Then in the function you need to substitute the value of the extreme points of this segment. There are two ways.
1) Having calculated the derivative and the intervals at which it is positive or negative, conclude that the function decreases on this segment or increases.
In accordance with them, substitute a larger or smaller value of the argument into the function.
2) Just substitute both points in the function and compare the obtained values of the function.
In which tasks is the derivative 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 top of the parabola is found by the formula.
If a <0, then the parabola branches are directed down. And its peak is the maximum point.
If a> 0, then the parabola branches are directed upward, the vertex is the minimum point.
After calculating the vertex point of the parabola, substitute its value in the function and calculate the corresponding value of the function.
2) The function y = tg x. Or y = ctg x.
These functions are monotonically increasing. Therefore, the larger 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 in a segment with examples.
Basic job types
Task: the largest or smallest function value. An example on the chart.
In the figure you see a 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 argument 2.
Answer: 2.
We continue to consider examples. Task: find the largest and smallest value of the function on the segment.
Find the smallest value of the function y = (x - 8) e x-7 on the segment [6; eight].
1. Take the derivative of a complex function.
y '(x) = (x - 8) e x-7 = (x - 8)' (e x-7 ) + (x - 8) (e x-7 ) '= 1 * (e x-7 ) + (x - 8) (e x-7 ) = (1 + x - 8) (e x-7 ) = (x - 7) (e x-7 )
2. Equate the resulting derivative to zero and solve the equation.
y '(x) = 0
(x - 7) (e x-7 ) = 0
x - 7 = 0, or e x-7 = 0
x = 7; e x-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) e 6-7 = -2e -1
y (7) = (7 - 8) e 7-7 = -1 * e 0 = -1 * 1 = -1
y (8) = (8 - 8) e 8-7 = 0 * e 1 = 0
The answer is -1.
So, in this article, the basic theory was examined on how to find the smallest value of a function on a segment, which is necessary for successfully solving exam questions in profile mathematics. Elements of mathematical analysis are also used in solving tasks from part C of the exam, but obviously they represent a different level of complexity, and their decision algorithms are difficult to fit into the framework of one material.