Function and the study of its features is one of the key chapters in modern mathematics. The main component of any function is graphs depicting not only its properties, but also the parameters of the derivative of this function. Let's take a look at this tricky topic. So what is the best way to find the maximum and minimum points of a function?
Function: Definition
Any variable that depends in some way on the values of another value can be called a function. For example, the function f(x2) is quadratic and determines the values for the entire set x. Let's say that x=9, then the value of our function will be equal to 92=81.
Functions come in many different types: logical, vector, logarithmic, trigonometric, numeric and others. Such outstanding minds as Lacroix, Lagrange, Leibniz and Bernoulli were engaged in their study. Their writings serve as a bulwark in modern ways of studying functions. Before finding the minimum points, it is very important to understand the very meaning of the function and its derivative.
The derivative and its role
All functions are independing on their variable values, which means that they can change their value at any time. On the graph, this will be depicted as a curve that either descends or rises along the y-axis (this is the whole set of "y" numbers along the vertical of the graph). And so definition of a point of a maximum and a minimum of function just is connected with these "oscillations". Let us explain what this relationship is.
The derivative of any function is drawn on a graph in order to study its main characteristics and calculate how quickly the function changes (ie changes its value depending on the variable "x"). At the moment when the function increases, the graph of its derivative will also increase, but at any second the function may begin to decrease, and then the graph of the derivative will decrease. Those points at which the derivative goes from minus to plus are called minimum points. In order to know how to find the minimum points, you should better understand the concept of the derivative.
How to calculate the derivative?
Defining and calculating the derivative of a function implies several concepts from differential calculus. In general, the very definition of the derivative can be expressed as follows: this is the value that shows the rate of change of the function.
Mathematical way to determine it for many students seems complicated, but in fact everything is much simpler. You just need to followstandard plan for finding the derivative of any function. The following describes how you can find the minimum point of a function without applying the rules of differentiation and without memorizing the table of derivatives.
- You can calculate the derivative of a function using a graph. To do this, you need to depict the function itself, then take one point on it (point A in Fig.) Draw a line vertically down to the abscissa axis (point x0), and at point A draw a tangent to function graphic. The abscissa axis and the tangent form an angle a. To calculate the value of how fast the function increases, you need to calculate the tangent of this angle a.
- It turns out that the tangent of the angle between the tangent and the direction of the x-axis is the derivative of the function in a small area with point A. This method is considered a geometric way to determine the derivative.
Methods of researching a function
In the school curriculum of mathematics, it is possible to find the minimum point of a function in two ways. We have already analyzed the first method using the graph, but how to determine the numerical value of the derivative? To do this, you will need to learn several formulas that describe the properties of the derivative and help convert variables like "x" into numbers. The following method is universal, so it can be applied to almost all kinds of functions (both geometric and logarithmic).
- It is necessary to equate the function to the derivative function, and then simplify the expression using the rulesdifferentiation.
- divide by zero).
- After that, you should convert the original form of the function into a simple equation, equating the entire expression to zero. For example, if the function looked like this: f(x)=2x3+38x, then according to the rules of differentiation, its derivative is equal to f'(x)=3x2 +1. Then we transform this expression into an equation of the following form: 3x2+1=0.
- After solving the equation and finding the points "x", you should draw them on the x-axis and determine whether the derivative in these areas between the marked points is positive or negative. After the designation, it will become clear at what point the function begins to decrease, that is, it changes sign from minus to the opposite. It is in this way that you can find both the minimum and maximum points.
Differentiation rules
The most basic part of learning a function and its derivative is knowing the rules of differentiation. Only with their help it is possible to transform cumbersome expressions and large complex functions. Let's get acquainted with them, there are quite a lot of them, but they are all very simple due to the regular properties of both power and logarithmic functions.
- The derivative of any constant is zero (f(x)=0). That is, the derivative f(x)=x5+ x - 160 will take the following form: f' (x)=5x4+1.
- The derivative of the sum of two terms: (f+w)'=f'w + fw'.
- Derivative of a logarithmic function: (logad)'=d/ln ad. This formula applies to all kinds of logarithms.
- Derivative of degree: (x)'=nxn-1. For example, (9x2)'=92x=18x.
- Derivative of a sinusoidal function: (sin a)'=cos a. If the sin of angle a is 0.5, then its derivative is √3/2.
Extremum points
We have already figured out how to find the minimum points, however, there is the concept of maximum points of a function. If the minimum denotes those points at which the function goes from minus to plus, then the maximum points are those points on the x-axis at which the derivative of the function changes from plus to the opposite - minus.
You can find the maximum points using the method described above, only it should be taken into account that they denote those areas where the function begins to decrease, that is, the derivative will be less than zero.
In mathematics, it is customary to generalize both concepts, replacing them with the phrase "extremum points". When the task asks to determine these points, this means that it is necessary to calculate the derivative of this function and find the minimum and maximum points.
