How to determine the monotonicity of a function 5

1. Definition Let x1 and x2 be any two numbers on the domain defined by the function f(x), and x1 x2, if f(x1) f(x2), then this function is an increasing function; On the contrary, if f(x1) f(x2), then this function is a subtraction function.

2. Nature Law.

In addition to using the monotonicity of basic elementary functions, the problem can also be simplified by using the relevant properties of monotonicity. If the functions f(x) and g(x) have monotonicity in interval b, then in interval b, they have: f(x) has the same monotonicity as f(x) c (c is constant);

f(x) is the same monotonicity as c f(x) when c 0 has the same monotonicity and when c 0 has the opposite monotonicity;

When f(x) and g(x) are both increase (decrease) functions, then f(x) and g(x) are both increase (decrease) functions; When f(x) and g(x) are both increase (decrease) functions, then f(x) g(x) is also an increase (decrease) function when both are greater than 0, and when both are constant less than 0, it is also a decrease (increase) function;

3. Derivative method.

A derivative less than 0 is decreasing, and an increase greater than 0 is equal to 0, which is the extreme point of the inflection point Common Methods for Finding the Range of Functions 1 Observation Method Used for simple analytic formulas. y 1 x 1, range ( 1] y = (1+x) (1-x)=2 (1-x)-1≠-1, range ( 1) (1, 2The matching method is mostly used for quadratic (type) functions.

y x 2-4x+3=(x-2) 2-1 -1, range [-1,

Common Problem Solving Methods:

Take either x1 > x2 on the defined field

Then bring x1 and x2 into the function to determine the size of f(x1) and f(x2).

If f(x1) is large, then it is an increasing function, and if f(x2) is large, then it is a decreasing function.

If there is an image to judge, the rising part of the function is the increasing function, and the falling part of the function is the decreasing function.

First, the derivative of the function is obtained, so that the derivative function is equal to zero, the value of x is obtained, and the relationship between x and the derivative function is judged.

In general, it cannot be seen at a glance... Except for easier questions.

The general method is to sit on the errand, sit on the quotient. And the others, I forgot.

In general, we let the unknowns x1 and x2 and compare the functions f(x1) and f(x2).

The composite functions are compared in their respective ranges and then the endpoint values of the intervals are compared.

You can see at a glance that I'm going to call you senior!

Draw a diagram or ask for a derivation.

The composite function is basically to disassemble and judge separately or draw a diagram to find a derivative.

I'll add that except for the primary function and the quadratic function, which are easier to see in high school, the others seem unlikely.

Kindness... Don't keep thinking about shortcuts! If you want to find me, I can summarize the monotonic finding method of function types one by one.

Will you ask for guidance? Giving a derivative of a function is the best way to judge the monotonicity of a function!

Find the derivative! The child picked mine.

No, look at whether it's up or down.

There are three ways to determine the monotonicity of a function:

1.Difference method (definition method).

According to the definition of increasing function and subtracting function, the monotonicity of the function is proved by the difference method, and the steps are: taking the value, making the difference, deforming, judging the number, and qualitative. Among them, the deformation step is the difficulty, and the common techniques are:

The integer type --- factorization and matching method, as well as the six-term formula method, the fractional type --- the merger and merge into a commercial formula, and the quadratic radical type --- the molecule is rationalized.

Specifically: first take two values on the interval, generally x1 and x2, set x1 x2 (or x1 x2) and then substitute x1 and x2 into the f(x) analytic formula to make the difference, that is, to calculate f(x1)-f(x2) The key step is to simplify, generally into the form of multiplication or division.

For example, if you set the condition of x1 x2 and finally simplify it to f(x1)-f(x2) 0, it is an increasing function in the interval and a decreasing function in the interval.

2.Image method.

The monotonicity of the function is judged by the continuous rise or fall of the function image.

3.Derivative method.

The monotonicity of the discriminant function is determined by the sign of the derivative function.

Definition of function monotonicity

In general, let the function definition domain be iIf for any two independent variables x1 and x2 on an interval d in the defined domain i, when x1 < x2, there is f(x1).< f(x2), then the function f(x) is said to be an increasing function over the interval d.

The monotonicity of a function can also be called the addition or decrease of a function.

Methods: 1. Image observation method.

As mentioned above, on the monotonic interval, the image of the increasing function is upward, and the image of the decreasing function is decreasing. Therefore, in a certain interval, the function corresponding to the function image that has been rising increases monotonically in that interval; The function image that has been decreasing corresponds to a monotonically decreasing function in that interval.

2. Derivative method.

Derivatives are closely related to the monotonicity of functions. It is another way to study functions, opening up many new avenues for it. Especially for specific functions, the use of derivatives to solve the monotonicity of the function is clear, the steps are clear, it is fast and easy to master, and the use of derivatives to solve the monotonicity of the function requires proficiency in the basic derivative formula.

If the function y=f(x) is derivable (differentiable) in the interval d, if there is always f at x d'(x)>0, then the function y=f(x) increases monotonically in the interval d; Conversely, if x d, f'(x) <0, then the function y=f(x) is said to decrease monotonically in the interval d.

The easiest way: derivative, the first derivative to find the highest point or the lowest point, the second derivative to determine whether to increase or decrease, the third year of high school textbooks have, read it yourself.

1) Definition method: According to the increase function, the definition of the subtraction function is judged according to "taking the value - making the difference - deforming - judging the symbol - drawing conclusions".

2) Image method: It is to draw an image of the function and judge the monotonicity of the function according to the rise or fall of the image.

2) Direct method: It is for the functions we are familiar with, such as primary function, quadratic function, inverse proportional function, etc.

Write out their monotonic intervals directly.

Let's give you a demonstration of how to solve the problem.

It is known that f(x)=-3x

Ask for his monotony on r.

Solution: Let x1, x2 r

and x1f:(x1)-f(x2)=(-3x2.)

1)-(3x1

3(x1-x2)

x1x1-x2<0

f(x2) The function is subtractive on r.

Well, this is the most common way to determine monotonicity and intervals.

To determine the monotonic interval depends on the topic.

with absolute values.

Example. y=|x

x-3|When x = 3 or -3.

The absolute values are 0

So there are 3 zones.

They are (- 3] and (-3, 3] and (3,).

2.Like the ones with the root number.

Recipe under the root number.

Then find the corresponding section.

3.Then there are some very common functions.

The monotonic interval of the primary function is the whole real number.

In the second case, you need to find the axis of symmetry (what it looks like when it is divided into two halves).

Inverse proportional function.

Generally (- 0) and (0,).

The monotonicity of functions is one of the important properties of functions, and it is usually discussed in the definition method, the image method, the composite function method, etc.

Increase + Increase = Increase, Decrease + Decrease = Decrease, Increase-Decrease = Increase, Decrease-Increase = Decrease, e.g

Let the function y f(x) be incremented on and a and b be constants

1) If a 0, then the function b af(x) is incremented on i;

2) If a 0, then the function b af(x) decrements on i

i.e. determine f(x1)-f(x2) (where x1 and x2 belong to the defined domain, assuming x1f(x2)

3. As shown on the right in the figure above, for this particular function f(x), we do not say that it is an increasing or decreasing function, but we can say that it is in an interval.

x1, x2].

Second, the nature of the operation.

1. F(x) has the same monotonicity as F(x)+a; f(x) with. g(x)

a·f(x) in.

a>0 has the same monotonicity when.

a<0, it has the opposite monotonicity;

2. When f(x) and g(x) are both increase (decrease) functions, if both are evergreen to zero, then f(x) g(x) is an increase (decrease) function; If both are consistently less than zero, it is a decrease (increase) function;

3. The sum of the two increasing functions is still the increasing function; The increase function minus the subtraction function is the increase function; The sum of the two subtraction functions is still a subtraction; The subtraction function minus the increase function is the subtraction function; When the value of the function is the same sign in the interval, the reciprocal of the increase (decrease) function is the decrease (increase) function.

The monotonicity of composite functions is "the same increases and the difference decreases". The specific connotation is that if the analytic expression of a composite function is y=f(u(x)), then its outer function is y=f(u) and the inner function is u=u(x).

1) If the monotonicity of the outer function y=f(u) with u as a variable and the inner function with x as a variable are the same (same increase or decrease), then y=f(u(x)) is the increasing function on this interval.

2) If the monotonicity of the outer function y=f(u) with u as a variable and the inner function with x as variable are opposite ("inner increase and outer subtraction" or "inner subtraction and outer increase") in an interval, then y=f(u(x)) is the subtraction function on this interval.

The increase or decrease of the above composite function can be simplified into the four cases shown in the following figure with mathematical formulas and symbols

Let the domain of the function y=f(u) be the domain of the god book du and the range of mu and the domain of the function u=g(x) be dx and the range of mx, if mx du ≠ then for any x in mx du pass u; If there is a uniquely determined value of y, then there is a functional relationship between the variable x and the zixun y through the variable u.

This function is called a composite function and is denoted as: y=f[g(x)], where x is called the independent variable, u is the intermediate variable, and y is the dependent variable (i.e., the function).

There are two ways to find monotonic intervals.

1. Derivative method: the derivative is less than 0 is decreasing, and more than 0 is increasing, equal to 0, which is the extreme point of the inflection point.

Firstly, according to the characteristics of the function image, the definition of the image language is obtained, if in a certain interval of the defined domain, the image of the function rises from left to right, then the function is an increasing function. If, in an interval of the defined domain, the image of the function descends from left to right, then the function is a subtraction function.

2. Definition method: set x1 and x2 to calculate (f(x1)-f(x2)) x1-x2), greater than 0 is increasing, less than 0 decreasing.

Secondly, if y increases with the increase of x in a certain interval, y is said to be an increasing function on the interval, and the interval is called the increasing interval of the function. If y decreases with the increase of x in an interval in an interval, then y is said to be a decreasing function on the interval, and the interval is called the decreasing interval of the function.

Application of function monotonicity.

1. Use the monotonicity of the function to find the maximum.

There are many ways to find the maximum (small) value of a function, but the basic method is to determine the monotonicity of the function, especially for the analysis of the maximum (small) value in the open or infinite region of the small derivable continuous point, which is generally determined by monotonicity.

2. Use the monotonicity of functions to solve equations.

Function monotonicity is a very important property of the function, because the monotonic function x and y are one correspondence, so that we can transform the miscellaneous equation into a form such as the "" equation through appropriate deformation, so as to use the function monotonicity to solve the equation x=a, so that the problem is simplified, and the construction of monotonic function is the key to solve the problem.