103 LearnersLast updated on July 9th, 2025
Increasing and Decreasing Intervals
Walking downstairs is easier than climbing up, just like how mathematical functions behave in calculus. In calculus, a function is considered increasing when its value rises with increasing x, and decreasing when its value drops as x increases. To verify the change in functions, we need to find the derivatives of such functions. A function is increasing if the function’s value goes up as x increases. A function is decreasing if the function’s value goes down as x increases.
What Are Increasing and Decreasing Intervals?
The increasing or decreasing intervals of a function show where its graph rises or falls. We can find these intervals by checking the sign of the first derivative in each interval.
- If the first derivative is positive on an interval, the function is increasing there.
- If the first derivative is negative, the function is decreasing on that interval.
How to Write Intervals of Increase and Decrease?
To write intervals of increase and decrease, you can follow these basic mathematical rules:
- An interval is said to be increasing if, for every x < y, the function satisfies f(x) ≤ f(y) for a real-valued function f(x).
- An interval is decreasing if, for every x < y, the function satisfies f(x) ≥ f(y).
You can also use the first derivative to identify these intervals:
- If the first derivative f'(x) ≥ 0 on an interval, the function is increasing there.
- If f'(x) ≤ 0, the function is decreasing on that interval.
This method makes it easier to analyze the behavior of functions and represent their increasing or decreasing nature.
How to Find Increasing and Decreasing Intervals?
Now that we understand what increasing and decreasing intervals mean, let’s learn how to find them for a function. We’ll use an example to understand the process step by step.
Consider the function:
f(x) = x³ + 3x² – 45x + 9
Step 1: Differentiate the function
Find the first derivative, f '(x):
f '(x) = 3x² + 6x – 45
= 3(x² + 2x – 15)
= 3(x + 5)(x – 3)
Step 2: Find critical points
Set f '(x) = 0:
3(x + 5)(x – 3) = 0
⇒ x = –5 and x = 3
Here, -5 and 3 are critical points. These are the points where the slope of the function changes.
Step 3: Identify intervals
Using the critical points, divide the number line into three intervals:
(–∞, –5), (–5, 3), and (3, ∞)
|
Interval |
Value of X |
f '(x) |
Increasing/ Decreasing |
|
(–∞, –5) |
x = –7 |
f '(–7) = 3(–2)(–10) = 60 > 0 |
Increasing |
|
(–5, 3) |
x = –2 |
f '(–2) = 3(3)(–5) = –45 < 0 |
Decreasing |
|
(3, ∞) |
x = 5 |
f '(5) = 3(10)(2) = 60 > 0 |
Increasing |
As a result, f(x) = x3 + 3x2 - 45x + 9 has increasing intervals (-∞, -5) and (3, ∞) and decreasing intervals (-5, 3).
What Are Critical Points and Extrema?
A critical point of a function occurs when the first derivative of the function is either zero or undefined. These points are important for identifying extrema, which are the local maximum or minimum values of the function.
Critical Point: A point x = c is a critical point if either f '(c) = 0 or f '(c) is undefined.
Extrema: If the function changes from increasing to decreasing or vice versa around a critical point, then it is called an extrema — either a local maximum or a local minimum.
Let's examine a graph of a curve. The curve will turn at certain points, either from increasing to decreasing or the other way around. These points are called extrema. They may be:
- Local Maximum: The highest point in a small surrounding area.
- Local Minimum: The lowest point within a certain small area.
How to Identify Increasing and Decreasing Intervals?
It becomes clear from the above figures that every extrema of a function is a point where its derivative changes sign. In other words, the function either goes from increasing to decreasing or vice versa. When identifying regions where a function is increasing or decreasing, it is important to examine the behavior around the extrema.
To find these intervals for any function f(x) over a given interval, follow these steps:
- Check if the function is differentiable and continuous in the given interval.
- Solve the equation f '(x) = 0. The solutions give the critical points (possible extrema).
- For each critical point x = c, examine the sign of the derivative f '(x) on either side of c to determine whether the function is increasing or decreasing in those intervals.
How to Check Continuity and Differentiability?
For a function to have increasing or decreasing intervals, it must be continuous and differentiable in that interval.
- Continuity means the graph of the function has no breaks or gaps at a point.
- Differentiability means the function’s first derivative exists at that point.
Every differentiable function is continuous, but a continuous function may not be differentiable.
Increasing and Decreasing Intervals Using Graph
In this section, we will learn how to visually represent increasing and decreasing intervals on a graph. This helps you identify these intervals simply by looking at the graph.
Below are two sample graphs of different functions. The first graph shows an increasing function because the curve goes up as we move from left to right along the x-axis. The second graph shows a decreasing function because the curve goes down as we move from left to right.
Real-Life Applications of Increasing and Decreasing Intervals
Increasing and decreasing intervals have real life applications in different fields. Here are few examples of applications -
- Businesses utilize these intervals to monitor their earnings and determine when to increase or reduce output. For example, if profit rises with the number of items sold, the interval is increasing. If profit rises in proportion to the quantity sold, the function is increasing. If profit declines due to increased costs, the function is decreasing.
- In the health sector, this concept helps to track the effectiveness of a medicine over time. An interval that is increasing would indicate higher effectiveness, while one that is decreasing indicates less efficacy.
- Governments can use these intervals to examine population growth. This helps analyze whether population growth is increasing or decreasing. It enables them to plan for social welfare.
Common Mistakes and How to Avoid Them in Increasing and Decreasing Intervals
Understanding increasing and decreasing intervals is an essential algebra skill. However, students often make mistakes when learning this concept. Here are a few common mistakes and ways to avoid them:
Mistake 1
Usage of Closed Intervals Instead of Open Intervals
Students might mistakenly use closed intervals [a, b] instead of open intervals (a, b).
Always use open intervals (a, b) because we cannot assure that the function will behave in an increasing or decreasing manner at the endpoints.
Mistake 2
Confusion between Slope and Value
Some students think a function is increasing just because its y-values f(x) are positive.
Verify not only if the values of f(x) are positive, but also whether they are increasing as x increases.
For example:
Consider f(x) = -x.
f(1) = −1, f(2) = −2, and f(3) = −3 (incorrect).
The function is decreasing because its values are getting smaller, even though they are negative.
Mistake 3
Overlooking Domain Restrictions
Students incorporate intervals in which the function is not defined.
Always ensure that you check the domain before writing intervals of increase or decrease.
For example:
Consider the function:
f(x) = x² + 2x + 1
This is a polynomial function, so it is defined for all real values of x.
There are no undefined points to exclude.
Steps to find increasing/decreasing intervals:
Differentiate the function:
f '(x) = 2x + 2
Find critical points by setting f '(x) = 0:
2x + 2 = 0
x = -1
Test intervals around x = -1:
For x < -1, f '(x) < 0 → function is decreasing
For x > -1, f '(x) > 0 → function is increasing
Final Answer:
The function is decreasing on (−∞, -1)
The function is increasing on (−1, ∞)
Mistake 4
Believing Every Critical Point is a Maxima or Minima
Students mistakenly believe the function must have a peak (maximum) or a dip (minimum) at f′(x) = 0.
Solution: Verify the sign of the derivative before and after the point. It is not a turning point if the sign stays the same.
For example:
Let f(x) = x³
f′(x) = 3x²
At x = 0, f′(0) = 0
However, f′(x) is positive on both sides of 0. Therefore, the function is neither a maximum nor a minimum; it simply keeps increasing.
Mistake 5
Inaccurate Interval Notation
Some students either write intervals incorrectly or confuse parentheses and brackets.
Write intervals from left to right (small to large) at all times. To describe increasing or decreasing intervals, use correct interval notation.
For example:
Correct: (−∞, 1)
[−∞, 1] or (1, −∞) are incorrect.
Solved Examples of Increasing and Decreasing Intervals
Problem 1
Determine the intervals where the function f(x) = x² increases and decreases.
Increasing on: (0, ∞)
Decreasing on: (−∞, 0)
Explanation
We first find the derivative → f′(x) = 2x
Set f′(x) = 0 → 2x = 0 → x = 0 (critical point)
Test sign of f′(x) around x = 0
f′(−1) = −2 → negative → decreasing
f′(1) = 2 → positive → increasing
So, f(x) = x2 is:
Increasing on: (0, ∞)
Decreasing on: (−∞, 0)
Problem 2
For the function f(x) = x³, determine the intervals of increase and decrease.
Increasing on: (−∞, ∞)
Decreasing: None.
Explanation
Let f′(x) = 3x²
Set f′(x) = 0 → 3x² = 0 → x = 0
We now test f′(x) around x = 0
f′(−1) = 3 → positive
f′(1) = 3 → positive
So, the f(x) = x3 is:
Increasing on: (−∞, ∞)
Decreasing: None.
Problem 3
Find where the function f(x) = sin(x) is increasing or decreasing in the interval [0, 2π].
Increasing on: (0, π/2) ∪ (3π/2, 2π)
Decreasing on: (π/2, 3π/2)
Explanation
Let f′(x) = cos(x)
We need to set f′(x) = 0 → cos(x) = 0 → x = π/2, 3π/2
Test the sign in intervals:
(0, π/2): cos(x) > 0 → increasing
(π/2, 3π/2): cos(x) < 0 → decreasing
(3π/2, 2π): cos(x) > 0 → increasing
So, the f(x) = sin(x) is:
Increasing on: (0, π/2) ∪ (3π/2, 2π)
Decreasing on: (π/2, 3π/2)
Problem 4
Find the increasing/decreasing intervals for f(x) = √x on (0, ∞)
Increasing on: (0, ∞)
Decreasing: None
Explanation
Let f(x) = x(1/2)
We then need to set f′(x) = (1/2)x(−1/2) = 1 / (2√x)
Since √x > 0 for all x > 0, f′(x) > 0 for all x > 0
So, the f(x) = √x is:
Increasing on: (0, ∞)
Decreasing: None
FAQs on Increasing and Decreasing Intervals
1.What do intervals of increasing and decreasing mean?
2.What are the signs that a function is increasing or decreasing?
3.What is the significance of increasing and decreasing intervals?
4.Is it possible for a function to simultaneously increase and decrease?
5.Do we write intervals with endpoints included?
6.How does learning Algebra help students in Australia make better decisions in daily life?
7.How can cultural or local activities in Australia support learning Algebra topics such as Increasing and Decreasing Intervals ?
8.How do technology and digital tools in Australia support learning Algebra and Increasing and Decreasing Intervals ?
9.Does learning Algebra support future career opportunities for students in Australia?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
