Monotone Increasing — Definition, Formula & Examples

Monotone increasing describes a function whose output either stays the same or rises as the input moves from left to right. If the output strictly rises (never stays the same), the function is called strictly increasing.

A function $f$ is monotone increasing (also called non-decreasing) on an interval $I$ if for all $a, b \in I$ with $a < b$ , it holds that $f(a) \leq f(b)$ . If the inequality is strict, $f(a) < f(b)$ , then $f$ is strictly increasing on $I$ .

Key Formula

a < b \implies f(a) \leq f(b)

Where:

$a, b$ = Any two input values in the interval, with a less than b
$f$ = The function being tested for monotone increase

How It Works

To check whether a function is monotone increasing on an interval, pick any two input values

a

and

b

in that interval where

a < b

and verify that

f(a) \leq f(b)

. In calculus, you can use the first derivative: if

f'(x) \geq 0

for every

x

in an interval, then

f

is monotone increasing there. If

f'(x) > 0

throughout the interval,

f

is strictly increasing. Graphically, a monotone increasing function never moves downward as you trace it from left to right.

Worked Example

Problem: Determine the interval(s) on which

f(x) = x^3

is monotone increasing.

Step 1: Find the derivative of

f(x)

f'(x) = 3x^2

Step 2: Determine where

f'(x) \geq 0

. Since

3x^2 \geq 0

for all real

x

, the derivative is non-negative everywhere.

3x^2 \geq 0 \quad \text{for all } x \in \mathbb{R}

Step 3: Note that

f'(x) = 0

only at

x = 0

, a single point. Everywhere else the derivative is strictly positive, so

f

is strictly increasing on

(-\infty, \infty)

Answer:

f(x) = x^3

is monotone (strictly) increasing on

(-\infty, \infty)

Why It Matters

Identifying where a function increases is essential for finding extrema, sketching curves, and solving optimization problems in calculus. In precalculus, recognizing monotone behavior helps you determine whether a function has an inverse, since a strictly increasing function is always one-to-one.

Common Mistakes

Mistake: Confusing "monotone increasing" (allows

f(a) = f(b)

) with "strictly increasing" (requires

f(a) < f(b)

Correction: Monotone increasing permits flat sections where the function value does not change. Strictly increasing means the output must grow whenever the input grows. A constant segment breaks strict increase but not monotone increase.