How to tell if a function is monotonicity

The definition of function monotonicity is that if the function y=f(x) is an increasing or decreasing function in a certain interval, then the function y=f(x) has strict monotonicity in this interval.

Note: The monotonicity of a function is also called the increase or decrease of a function.

Steps to judge:

a.Let x1 and x2 belong to the given interval, and x1bCalculate f(x1)-f(x2) to the simplest.

c.Determine the sign of the above difference.

d.Draw conclusions (if the difference <0 is the increasing function, if the difference is 0, then it is the subtraction function).

Monotonicity is for an interval, y=x squared + 1 is decreasing on the left side of the coordinate axis and increasing on the right side. It does not have an increase or decrease in the strict sense.

You have to pay attention to the fact that monotonicity is for an interval in the definition domain, it is a local concept, and some functions are incrementing and some are decreasing in the definition domain.

You judge whether the given function is monotonicity in its definition domain, just look at whether the function is monotonic in the whole definition domain or a certain interval in a given definition domain, to put it bluntly, it cannot increase or decrease.

Can you see it?

You can see it by drawing the function graph.

y = x squared + 1, which is a quadratic function, its image is symmetric with respect to the y axis, in (0, negative infinity) the function is decreasing, (0, positive infinity) is incremental. It is that it is tonal in these two intervals. But the whole domain of definitions (negative, positive infinity) cannot be said to be monotonous.

If the function y=f(x) is an increasing or decreasing function in a certain interval, then the function is said to have (strict) monotonicity in this interval, and this interval is called the monotonic interval of the function. In this case, it is also said that the function is a monotonic function on this interval.

What is it?

Hit it out and take a look.

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.

Derivative, in the reciprocal interval greater than 0 is an increase and vice versa, or you can also use the definition of monotonicity.

The nature of the exponential function.

1. Define the domain: r

2. Value range: (0,

3. Passing the point (0, 1), that is, when x=0, y=1

4. When a 1, it is an increasing function on r; When 0 a 1, on r is a subtractive function.

5. The function graphs are all concave.

6. Functions always tend infinitely towards the x-axis in a certain direction and never intersect.

7. The exponential function is unbounded.

8. Exponential functions are non-odd and non-even functions.

The derivative, f, is generally used'(x) > Dounian 0, the derivative function f(x) increases monotonically, f'(x) <0, monotonically decreasing, stool clear.

Of course, common functions and conformances can be observed to determine the empty rough difficulty, such as increase function + increase function = increase function, increase function - subtraction function = increase function. Take the reciprocal, or multiply by a negative value, and the monotonicity is reversed.

Step 1: Derive the function.

Step 2: Let the derivative function be greater than 0, and find the value range of x as the increasing range of the function.

If the derivative function is less than 0, the range of x is the decreasing interval of the function.