What is the definition of a monotonic function?

What is the concept of monotonic function

Generally, let the domain of function f(x) be I:

If for the values x1 and x2 of any two independent variables in an interval within I, when x1

If for the values x1 and x2 of any two independent variables belonging to a certain interval in I, when x1f(x2). Then f(x) is a decreasing function in this interval.

If the function y=f(x) is an increasing or decreasing function in a certain interval, it can be said that the function y=f(x) has monotonicity in that interval. This interval is called the monotonic interval of the function y=f(x). On a monotonic interval, the graph of an increasing function is rising, and the graph of a decreasing function is descending.

Note: (1) The monotonicity of a function is also called the increase or decrease of a function;

(2) The monotonicity of a function is for a certain interval, and it is a local concept;

(3) Method steps for determining the monotonicity of a function on a certain interval:

a. Let x1, x2∈ given interval, and x1

b. Calculate f(x1)-f(x2) to its simplest form.

c. Determine the sign of the above difference.

What is monotony

is a monotonic function

Generally, let the domain of function f(x) be I:

If for the values x1 and x2 of any two independent variables belonging to a certain interval in I, when x1

If for the values x1 and x2 of any two independent variables belonging to a certain interval in I, when x1 f(x2). Then f(x) is a decreasing function in this interval.

If the function y=f(x) is an increasing or decreasing function in a certain interval. Then it is said that the function y=f(x) has (strict) monotonicity in this interval. This interval is called the monotonic interval of y= f(x). The graph of the increasing function on the monotonic interval is rising, and the graph of the decreasing function is rising. The graph of the function is descending.

Note: (1) The monotonicity of a function is also called the increase or decrease of a function;

(2) The monotonicity of a function is for a certain interval, and it is a local concept;

(3) There are two main methods to determine the monotonicity of a function on a certain interval:

1) Definition method

a. Suppose x1, x2∈ given interval, and x1

b. Calculate f(x1)-f(x2) to its simplest form.

c. Determine the sign of the above difference.

2) Introduction method

Use the derivative formula to conduct the derivative, and then judge the relationship between the derivative function and 0 to determine the increase or decrease. The value of the derivative function is greater than 0, indicating that it is an increasing function. The value of the derivative function is less than 0, indicating that it is a decreasing function. The premise is that the original The function must be continuous.

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