This is a beta course, so its structure, chapters, and examples may continue to change.

Monotonicity of Functions

Definition

Monotonicity of a Function

Let $f(x)$ be defined on an interval $I$ :

Strictly increasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) < f(x_2)$
Increasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) \leq f(x_2)$
Strictly decreasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) > f(x_2)$
Decreasing: for any $x_1, x_2 \in I$ , if $x_1 < x_2$ then $f(x_1) \geq f(x_2)$

Definition method: use the definition directly
Derivative method: for differentiable functions, decide monotonicity from the sign of the derivative
- If $f'(x) > 0$ , then $f(x)$ is strictly increasing
- If $f'(x) < 0$ , then $f(x)$ is strictly decreasing
- If $f'(x) \geq 0$ , then $f(x)$ is increasing (non-decreasing)
- If $f'(x) \leq 0$ , then $f(x)$ is decreasing (non-increasing)
Graphical method: read monotonicity from the graph

The derivative method may be unfamiliar—that is fine; you will meet it when you study calculus.

Determine the monotonicity of $f(x) = x^3 - 3x + 1$ on $\mathbb{R}$ .

Answer and Explanation(4 个标签)

monotonicityderivativefunction graphcubic function

Approach: Use the derivative to decide monotonicity.

Detailed steps:

Differentiate: $f'(x) = 3x^2 - 3 = 3(x^2 - 1) = 3(x-1)(x+1)$
Analyze the sign of the derivative:
- When $x < -1$ , $f'(x) > 0$ , so $f$ is strictly increasing
- When $-1 < x < 1$ , $f'(x) < 0$ , so $f$ is strictly decreasing
- When $x > 1$ , $f'(x) > 0$ , so $f$ is strictly increasing

Answer: The function is strictly increasing on $(-\infty, -1]$ and $[1, +\infty)$ , and strictly decreasing on $[-1, 1]$ .

Determine the monotonicity of $f(x) = \frac{x^2 + 1}{x}$ on $(0, +\infty)$ .

Answer and Explanation(4 个标签)

monotonicityderivativefunction graphrational function

Approach: Use the derivative to decide monotonicity.

Detailed steps:

Differentiate: $f'(x) = \frac{2x \cdot x - (x^2 + 1) \cdot 1}{x^2} = \frac{2x^2 - x^2 - 1}{x^2} = \frac{x^2 - 1}{x^2}$
Analyze the sign of the derivative:
- When $0 < x < 1$ , $f'(x) < 0$ , so $f$ is strictly decreasing
- When $x > 1$ , $f'(x) > 0$ , so $f$ is strictly increasing

Answer: The function is strictly decreasing on $(0, 1]$ and strictly increasing on $[1, +\infty)$ .

Symbol	Type	Reading/Explanation	Meaning in This Article
$f(x)$	Math symbol	f of x	A function of variable $x$
$x_1, x_2$	Math symbol	x one, x two	Two points inside the interval
$f'(x)$	Math symbol	f prime of x	The first derivative of the function
$I$	Math symbol	I	An interval
$\mathbb{R}$	Math symbol	Real numbers	The set of all real numbers

English term	Chinese term	Phonetic	Explanation
monotonicity	单调性	/ˌmɒnəʊtəˈnɪsɪti/	How function values change as the input grows
monotonically increasing	单调递增	/ˌmɒnəʊˈtɒnɪkli ɪnˈkriːsɪŋ/	Values grow as the input grows
strictly monotonically increasing	严格单调递增	/ˈstrɪktli ˌmɒnəʊˈtɒnɪkli ɪnˈkriːsɪŋ/	Values strictly grow as the input grows
monotonically decreasing	单调递减	/ˌmɒnəʊˈtɒnɪkli dɪˈkriːsɪŋ/	Values shrink as the input grows
strictly monotonically decreasing	严格单调递减	/ˈstrɪktli ˌmɒnəʊˈtɒnɪkli dɪˈkriːsɪŋ/	Values strictly shrink as the input grows
derivative	导数	/dɪˈrɪvətɪv/	Rate of change of a function
inverse function	反函数	/ɪnˈvɜːs ˈfʌŋkʃən/	The inverse mapping of a function
one-to-one correspondence	一一对应	/wʌn tuː wʌn ˌkɒrɪˈspɒndəns/	Each input has one output and each output one input