# Increasing, Decreasing and Monotonic Functions
Source: algebrica.org - CC BY-NC 4.0
https://algebrica.org/increasing-and-decreasing-functions/
Fetched from algebrica.org post 8284; source modified 2025-12-06T17:55:43.
## Introduction
Understanding the behavior of [functions](../functions.md) is fundamental in mathematics. Depending on how their output values change with respect to the input, functions can be classified as:
- increasing
- decreasing
- monotonic
##### These characteristics are closely tied to the geometry of their graphs in the Cartesian plane, revealing whether a function rises, falls, or maintains a consistent directional trend.
* * *
Let \\( y = f(x)\\) be a function defined on a [domain](../determining-the-domain-of-a-function.md) \\( X \\subseteq \\mathbb{R} \\). We say that \\( f \\) is **strictly increasing** on an interval \\(I \\subseteq X \\) if, for any two values \\( x\_1, x\_2 \\in I \\) such that \\( x\_1 < x\_2 \\), the following condition holds:
\\[
f(x\_1) < f(x\_2)
\\]
This means that as the input \\( x \\) increases within the interval \\( I \\), the output \\( f(x) \\) also strictly increases, without any flat or decreasing segments.
* * *
Let \\( y = f(x) \\) be a function defined on a domain \\( X \\subseteq \\mathbb{R} \\). We say that \\( f \\) is **strictly decreasing** on an interval \\( I \\subseteq X \\) if, for any two values \\( x\_1, x\_2 \\in I \\) such that \\( x\_1 < x\_2 \\), the following condition holds:
\\[
f(x\_1) > f(x\_2)
\\]
This means that as the input \\( x \\) increases within the interval \\( I \\), the output \\( f(x) \\) strictly decreases, with no flat or increasing sections.
* * *
A function with domain \\( X \\subseteq \\mathbb{R} \\) is said to be **strictly monotonic** on an interval \\( I \\subseteq X \\) if it is either strictly increasing or strictly decreasing throughout the entire interval \\( I \\), with no change in direction or flat segments. In other words, the function maintains a consistent trend, either upward or downward, across \\( I \\).
To summarize: let \\( X \\subseteq \\mathbb{R} \\), and let \\( x\_1, x\_2 \\in X \\) with \\( x\_1 < x\_2 \\). Then the function \\( f : X \\to \\mathbb{R} \\) is said to be:
- Increasing: if \\( f(x\_1) \\leq f(x\_2) \\).
- Strictly increasing: if \\( f(x\_1) < f(x\_2) \\).
- Decreasing: if \\( f(x\_1) \\geq f(x\_2) \\).
- Strictly decreasing: if \\( f(x\_1) > f(x\_2) \\).
- (Strictly) monotonic: if the function is either (strictly) increasing or (strictly) decreasing.
## Derivatives and monotonic behavior
We know that [derivatives](../derivatives.md) are used to describe the shape and graph of functions. In particular, the first derivative of a function, \\( f’(x) \\), can indicate the intervals where the original function \\( f(x) \\) is increasing and where it is decreasing.
In general, given a function \\( y = f(x) \\) that is [continuous](../continuous-functions.md) on an interval \\( I \\) and differentiable at the interior points of \\( I \\):
- If \\( f’(x) > 0 \\) for every \\( x \\) in the interior of \\( I \\), then \\( f(x) \\) is increasing on \\( I \\).
- If \\( f’(x) < 0 \\) for every \\( x \\) in the interior of \\( I \\), then \\( f(x) \\) is decreasing on \\( I \\).
- If \\( f’(x) = 0 \\) for every \\( x \\) in the interior of \\( I \\), then \\( f(x) \\) is constant on \\( I \\).
* * *
To demonstrate these properties, we use the [Lagrange’s Theorem](../lagrange-theorem.md). Let’s imagine having two points \\( a \\) and \\( b \\) \\(\\in I \\) with \\( a < b \\). Next, let us consider a point \\( c \\) belonging to the interval \\( \]a, b\[ \\).
By the Lagrange’s Theorem, we have:
\\[
f^{\\prime}\\left (c \\right ) = \\frac{f(b) - f(a)}{b - a}
\\]
Since we have \\( b - a > 0 \\) and \\( f’\\left (c \\right) > 0 \\), it follows that \\( f(b) - f(a) > 0 \\), which implies \\( f(b) > f(a) \\). Since \\( a \\) and \\( b \\) are arbitrary points in \\( I \\), the function is increasing on \\( I \\).
Similarly, considering the opposite case, since we have \\( b - a > 0 \\) and \\( f’\\left( c \\right) < 0 \\), it follows that \\( f(b) - f(a) < 0 \\), which implies \\( f(b) < f(a) \\). Since \\( a \\) and \\( b \\) are arbitrary points in \\( I \\), the function is decreasing on \\( I \\).
## Example 1
Let us consider the function:
\\[
f(x) = \\frac{x^4}{4} - \\frac{x^2}{2}
\\]
* * *
Let us compute its derivative:
\\[
f’(x) = x(x^2 - 1)
\\]
Let us find the intervals where the derivative is greater than zero. We have:
\\[
x > 0
\\]
\\[
x^2 - 1 > 0 \\implies x < -1 \\text{ or } x > 1
\\]
By multiplying the signs of the first and second factors, we obtain the intervals where the derivative is positive.
\\[
-1
\\]
\\[
0
\\]
\\[
1
\\]
\\( x > 0 \\)
\\( \\boldsymbol{-} \\)
\\( \\boldsymbol{-} \\)
\\( \\boldsymbol{+} \\)
\\( \\boldsymbol{+} \\)
\\( x^2 - 1 > 0 \\)
\\( \\boldsymbol{+} \\)
\\( \\boldsymbol{-} \\)
\\( \\boldsymbol{-} \\)
\\( \\boldsymbol{+} \\)
\\[
f’(x)
\\]
\\( \\boldsymbol{-}\\)
\\( \\boldsymbol{+} \\)
\\( \\boldsymbol{-} \\)
\\(\\boldsymbol{+} \\)
Therefore, the derivative \\( x(x^2 - 1) \\) is positive for:
\\[
x \\in (-1,0) \\cup (1,+\\infty)
\\]
##### For the sake of completeness, we recall that the sign analysis of a function, as in the given example, requires examining the signs of its individual factors and determining the overall sign for each interval by computing the product of these signs.
* * *
Graphically, its behavior is as follows:
Therefore, the function is increasing in the interval \\((-1,0) \\cup (1,+\\infty)\\) and decreasing in the interval \\((-\\infty, -1) \\cup (0,1)\\).
