How to find the number of zeros in a function? How to find the number of zeros of a function

Let's look at monotonicity first, if you increase or decrease within the defined domain, you won't have to say, and the result of finding the minimum or maximum value will come out. If you increase or decrease in segments, find the derivative, get all the extreme points, calculate the coordinates, and then draw a monotonicity trend chart on the coordinate axis, and the number of zero points will be clear at a glance. I haven't touched any of this in years.

It's supposed to be high school, right? That's what it looks like in the image, I hope it can help you!

Use the number axis to penetrate the root method.

From top to bottom, from right to left, odd times and one wear, even times and one can't wear.

1. Use the solution equation to determine the number of zeros of the function.

Example 1: The number of zeros of the function f(x)=x2+2x-3,x 0,-2+lnx,x>0 is.

When x 0 is solved, x2+2x-3=0 is solved, and x=-3 is solvedWhen x>0, let -2+lnx=0, and x=e2So, the function f(x) has 2 zeros. I choose C

2. Use the function image to determine the number of zeros of the function.

1.Directly observe the number of intersections between the function image and the x-axis.

According to the definition of the zero point of the function, the image of the function y=f(x) can be made, and the number of points of intersection with the x-axis is the number of zeros of the function. This method is suitable for functions that are easy to make images.

2.The number of intersections that are divided into two function images.

The zero point of the function f(x)=f(x)-g(x), that is, the root of the equation f(x)=g(x), that is, the abscissa of the intersection of the image of the function y=f(x) and the image of the function y=g(x). When the image of the function y=f(x) is not easy to make, f(x) can be decomposed into two relatively simple functions, i.e., f(x)=f(x)-g(x), and the number of intersections of the image of f(x) and g(x) is used to determine the number of zeros of f(x).

Example 2 assigns that the function f(x) defined on r is an even function with a minimum positive period of 2 and f(x) is a derivative of f(x). When x [0, ], 0 solves when x (0, ) and x≠, (x-

f(x)=0 to find the number of zeros.

Method 1 makes y=f(x) and derives it to obtain the monotonicity of the function in each interval.

By observing the limits of the left and right ends of the defined domain, the left and right limits of discontinuous points, and the function values of each station, the number of zeros can be obtained by combining monotonicity.

For example, lnx 1 (x 1) = 0 fractions.

Let f(x)=lnx 1 (x 1).

The function is discontinuous at x=1.

f'(x)=1/x＋1/(x–1)²＞0

So the function is monotonically increasing at (0,1), (1, monotonically increasing lim(x 0) f(x)=

lim(x→1–) f(x)=＋∞

lim(x→1＋) f(x)=–∞

lim(x→＋∞f(x)=＋∞

According to monotonicity, the function f(x) must have a zero point on (0,1) and a zero point on (1,).

So f(x)=0 has two zeros.

The second method is to combine numbers and shapes to transform the zero problem into the intersection problem of two functions, and draw an image to obtain the number of intersection points by studying the properties of the two functions.

For example, lnx 1 (x 1)=0

lnx=1/(x–1)

It can be transformed into an intersection problem of f(x)=lnx and g(x)=1 (x 1), and the image can be drawn to obtain two intersections, that is, the original equation has two zero points.

is the value of the function . On the function image, it is the abscissa of the image and the intersection point.

So we can start from two aspects to find the zero point: find the solution of ; Find the image cross-section.

Let's take a look at what specific methods there are:

Solving equations: Solving equations to get the zero point;

Combination of numbers and shapes: This is a frequently used analysis method, especially in optional questions.

Zero-point existence theorem: Using the zero-point existence theorem to determine whether there is a zero point in a certain interval is an important method in solving problems;

Finding the number of zeros: When finding the number of zeros, it is necessary to determine the existence of each monotonic interval, and at the same time, the existence of the zero point of the monotonic interval.

In the process of solving specific problems, we will also encounter complex functions, first transform the complex problems into simple problems, and then choose the appropriate method to find the zero point.

Let's look at a specific example.

Example 1] (2018 National 2 volumes 21-2) known function , proving: There is only one zero point.

Analysis] is a cubic function with parameters, which seems to be a cubic function to find the number of zeros, but it becomes complicated with the parameters, so at this time, you can transform it and separate the parameters to find the number of solutions. This is further transformed into a zero-point problem for functions.

Analysis] because Heng is established. So the number of zeros is equivalent to the number of zeros of the function of the function.

To judge monotonicity first, use the derivative method: if and only if monotonically incremented. So there is at most one zero, and thus at most one zero.

And because there is exactly a zero point.

Summary] Readers should be able to understand the separation parameters, but why they chose is confused. This belongs to the content of finding points (the inner point theorem), and we will devote a special chapter to explain this content later. Let's first understand the application of the zero-point existence theorem.

In this section, we focus on explaining how to find the number of zeros, which is also a hot question type in the college entrance examination in recent years, and it is also the key problem we will face in the zero-point problem.

Example 2] (2019 National Volume 2 Rational Numbers 20-1 adaptation) The number of zeros of the known function is found.

Analysis] To find the number of zeros, we require the monotonic intervals of the function, and then judge the existence of the zeros of each monotonic interval.

Analysis] defines the domain as and is determined by the sum method: and are monotonically increasing on , so in is monotonically increasing;

Monotonically increasing on , when , when , by the zero point existence theorem and monotonicity, where there is a unique zero point,

Method 1: Definition.

Steps: The first step is to judge the monotonicity of the function;

The second step is to verify whether the product of the pure branch imaginary value of the function at the end of the interval is less than 0 according to the existence theorem of the zero point. If its product is less than 0, then the interval is the existence of a unique zero point interval or the zero point is calculated directly by using the idea of the equation;

Step 3 Draw conclusions.

Example].The number of zeros of the function is ( ).

a．0 b．1 c．2 d．3

Analysis] is known.

Therefore, in is monotonically increasing, and , so the number of zeros of is 1, so choose b

Fang do burning method 2: number combination method.

Problem solving steps: The first step is to convert a zero-point problem into an equation with a root;

Step 2 In the same Cartesian coordinate system.

, draw the images of the functions and , respectively;

Step 3 Observe and judge the number of intersections of the image of the function and .

Step 4: The number of intersections of the and images is equal to the zero point of the function.

Example].The number of solutions to the equation is ( ).

a．3 b．2 c．1 d．0

Analysis] From the image, it can be seen that the function and the function have 2 intersections, so the equation has 2 solutions, choose b

The zero point of the variable sign is that the image of the function passes through that point, that is, the value on both sides of the point is a different sign (the value of the function at that point is zero), in general, for the function y=f(x)(x r).

The real root x of the equation f(x)=0 is called the zero of the function y=f(x)(x r). That is, the zero point of the function is the value of the argument that makes the value of the function 0. The zero point of a function is not a point, but a real number.

1. Solve the value of the zero point: (1) Let the function f(x) be 0, and the value of the solution x is the zero point. (2) Divide the function to zero, split the function into two new functions, and then draw the approximate image of the two functions to judge the zero point by judging the intersection point of the two images.

The abscissa of the intersection is the zero point. The idea is to find the value of the argument that corresponds to the value of the function when it is zero.

2. The interval where the zero point is located: (1) When the question is a multiple-choice question, the answer endpoint value can be substituted into the function formula for evaluation, and when the function value satisfies one positive and one negative, that is, the interval where the value of the two functions is multiplied less than zero is the interval where the zero point is located. (2) Split the function into two functions, draw the images of the two functions, and then judge the intersection interval of the two function images through the image, which is the zero point interval.

3. Solve the number of zero points: find the derivative of the function, use the derivative function to find the increase and decrease interval of the blind sock out of the function, the maximum minimum and the maximum and minimum value (sometimes the maximum and minimum value of the function may be at the pole), and judge the number of intersection points between the function image and the horizontal axis (x=0) is the number of zero points. The idea is to find the number of intersections between the function and the horizontal axis.

There are three ways to find the zero point of a function:

1. Deform the function in an appropriate way (e.g. x2+5x+4). The higher terms (e.g., x2) are ranked in the first and lower terms in the back from left to right until the constant terms (e.g., 8 or 4). After the last item, add an equals sign and the number 0.

Arrange the correct polynomials:

x2 + 5x + 6 = 0

x2 - 2x – 3 = 0

Misarranged polynomials:

5x + 6 = x2

x2 = 2x + 3

2. Use letters a, b, c and so on to express the coefficients of the equation. This step does not require mathematical knowledge, and only reduces the difficulty of subsequent factorization through certain expressions. The equation you're trying to solve has a general form.

For the above equation, the general form is ax2 bx c = 0. All you need to do is find the number (coefficient) corresponding to the three letters in the equation you have arranged. For example:

x2 + 5x + 6 = 0

a = 1 (no number in front of "x" =1, as there is still one "x")

b = 5c = 6

x2 - 2x – 3 = 0

a = 1 (no number in front of "x" =1, as there is still one "x")

b = 2c = 3

3. Write down all the factor pairs of the constant term c. The factor pair of a number refers to the multiplication result of two numbers equal to the number. When writing because of the number of defeats, it is particularly dry or pay attention to negative numbers, and the multiplication of two negative numbers equals a positive number.

There is no strict requirement for the order of the two numbers in a factor pair (i.e., 1 4 is equivalent to 4 1).

Example: The factor pair of the constant term 6 in the equation x2 + 5x + 6 = 0 is:

1 x 6 = 6

1 x -6 = 6

2 x 3 = 6

2 x -3 = 6

How to determine the number of zeros of a function:

1. Let the value of the function be equal to zero, solve the equation, and the number of solutions is the number of zeros of the function.

2. Basic elementary functions take advantage of their properties. For example, quadratic functions are used with discriminant formulas.

3. Using the zero-point existence theorem: if the endpoint of the interval is different in the endpoint of the interval and the value of the round, the function has at least one zero point in this open interval.

4. Using the zero-point uniqueness theorem: the monotonic continuous function on the closed interval reduces the number of rounds, if the endpoint function value of the interval is different, then the function has a unique zero point in this open interval.

5. Note: If necessary, the derivative is used to judge monotonicity.

The even function f(x) on r satisfies f(x+2)=f(x), and when x belongs to 0,1, f(x)=x, then the function, i.e., y=x, and the even function f(x)=f(-x), then f(x)=|x|, are two straight lines with a slope of 1 over the origin and symmetry with respect to the origin;

function y=f(x)-log3 |x|, find the derivative y' = 1-( ln3 |x|), when x = ln3,y'=0, replace x=ln3 with y=f(x)-log3 |x|to get four coordinate points. There are 4 zeros.

You can first find out its axis of symmetry, and then draw a rough graph, and then have fx+2=fx, and you can draw an approximate curve of 2 to 3, and because it is an even function, you can also draw it from -3 to -2, and the graph is completely out, that's all. If you don't give points, I will only talk about the analysis process, and you can do the specific disintegration process yourself.