5.3 Determining Intervals on Which a Function is Increasing or Decreasing
What can the derivative of a function show us about the function itself? Can it tell us when the function increases or decreases? Yes, it can! In this lesson, we’ll delve into how we can use derivatives to determine when a function increases or decreases. 📈
🕑 When Does a Function Increase or Decrease?
In order to determine the intervals on which a function is increasing or decreasing, we first need to understand the concept of the derivative. The derivative of a function is the rate of change of the function at a given point.
Thus, we know the following:
- ➕=📈 If the derivative is positive at a certain point (which means the rate of change is positive at that point), the function is increasing at that point.
- ➖=📉 If the derivative is negative at a certain point, the function is decreasing at that point.
Take a look at this graph to see these trends in action. The gray line represents the function, , and the black line represents its derivative, .
Now, you may be thinking that finding the interval where a function is increasing or decreasing is as simple as finding the interval where the function’s derivative is positive or negative, respectively. Well, it is! 🪄
Where can a function change from increasing to decreasing and vice versa? It can only change its direction from increasing to decreasing and vice versa at its critical points, points where the function’s derivative equals or is undefined, and the points where the function itself is undefined.
So, for each of the intervals defined by the points where the function can change behavior, we can determine whether the function is increasing or decreasing on the interval by just plugging a point on that interval into the function’s derivative and seeing if the result is positive or negative. If it’s positive, then the function is increasing on that interval; if it’s negative, then the function is decreasing on that interval. Let’s give it a try!
✏️ Function Behavior: Walkthrough
➡️ Let be a function defined for all real numbers except Also let be defined as . On which intervals is increasing?
From what we learned above, we can analyze the intervals where is increasing by looking for the intervals where its derivative is positive.
A function can only change its direction from increasing to decreasing and vice versa at its critical points and the points where the function itself is undefined. Based on the problem statement, we determine that in this case, the only points where can change direction are and . We determine this by doing the following and solving:
at so it is a critical point and is undefined at as stated in the problem.
These two points divide the number line into three intervals:
Let’s evaluate at each interval to see if it’s positive or negative on that interval and therefore see if is increasing or decreasing on the interval. To evaluate on an interval, we can choose any x-value within the interval to substitute with and calculate.
In the table below, you’ll notice that we chose for the first interval, for the second interval, and for the third! Plug in, solve, and determine the behavior of the function.
| Interval | Verdict | ||
|---|---|---|---|
| is decreasing | |||
| is increasing | |||
| is increasing |
In conclusion, is increasing on the intervals and .
🪜 Steps to Determining Function Behavior
Here are the steps we used if you prefer seeing them in list form:
- 🎯 Determine the critical points of the function, where equals zero or is undefined.
- 🔢 Divide the function's domain into intervals based on the critical points determined. This is where you break the number line into pieces to see its behavior!
- 🔨 For each interval, choose a test point within that interval.
- ✏️ Evaluate the function's derivative at the chosen test point.
- 🏁 Interpret the result.
- If the result is positive, the function is increasing on that interval.
- If the result is negative, the function is decreasing on that interval.
Now you can put these steps into practice!
📝 Function Behavior: Practice Problems
Time to solve some problems! 🔍
❓ Function Behavior: Problems
Function Behavior: Question 1
Let .
On which interval(s) is decreasing?
Function Behavior: Question 2
Let .
On which interval(s) is increasing?
✅ Function Behavior: Answers and Solutions
Function Behavior: Question 1
The answer to this problem is . Let’s get into why. ⬇️
We can analyze the intervals where is decreasing by looking for the intervals where its derivative is negative.
A function can only change its direction from increasing to decreasing and vice versa at its critical points and the points where the function itself is undefined. So first, take the derivative and find the critical points.
Since the derivative of is , the only points where can change direction are and as at only these points.
These two points divide the number line into three intervals: .
Let’s evaluate at each interval to see if it’s positive or negative on that interval and therefore if is increasing or decreasing on the interval.
| Interval | Verdict | ||
|---|---|---|---|
| is increasing | |||
| is decreasing | |||
| is increasing |
In conclusion, is decreasing on the interval .
Function Behavior: Question 2
The answer to this problem is and . Here’s why:
We can analyze the intervals where is increasing by looking for the intervals where its derivative is positive.
A function can only change its direction from increasing to decreasing and vice versa at its critical points and the points where the function itself is undefined.
Since the derivative of is , which we can rewrite as , the only points where can change direction are , , and as at only these points.
These three points divide the number line into four intervals: .
Let’s evaluate at each interval to see if it’s positive or negative on that interval and therefore if is increasing or decreasing on the interval.
| Interval | Verdict | ||
|---|---|---|---|
| is decreasing | |||
| is increasing | |||
| is decreasing | |||
| is increasing |
In conclusion, is decreasing on the intervals and .
⭐ Closing
Great work! You now have the steps and practice to determine where a function changes from increasing to decreasing.
Vocabulary
The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.
| Term | Definition |
|---|---|
| decreasing | An interval on which a function's output values are getting smaller as the input values increase, corresponding to where the first derivative is negative. |
| first derivative | The derivative of a function, denoted f', which describes the rate of change and indicates where a function is increasing or decreasing. |
| increasing | An interval on which a function's output values are getting larger as the input values increase, corresponding to where the first derivative is positive. |
Frequently Asked Questions
How do I find where a function is increasing or decreasing using derivatives?
You use f′ to tell where f goes up or down. Steps: 1. Find f′(x). Solve f′(x) = 0 and note where f′ is undefined—those are critical points (including endpoints if on a closed interval). (Keywords: first derivative, critical point, stationary point.) 2. Make a sign chart: break the domain into intervals at those critical points. Pick one test point in each interval and evaluate f′ there. If f′(test) > 0, f is increasing on that interval; if f′(test) < 0, f is decreasing. (This is the First Derivative Test idea from the CED: FUN-4.A.1.) 3. Use endpoints and nondifferentiable points as needed (they can affect local/global behavior). If you need to justify local maxima/minima, check sign change of f′ around the critical point. On the AP exam you’ll be asked to justify intervals using f′ (show derivative work, critical points, sign chart or test points). For a handy walk-through and examples, see the Topic 5.3 study guide (Fiveable) here: (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y). For extra practice, use the AP Calc practice bank (https://library.fiveable.me/practice/ap-calculus).
What's the formula for determining if a function is increasing or decreasing?
The rule is simple: use the first derivative. If f′(x) > 0 on an interval, f is increasing there; if f′(x) < 0, f is decreasing there. Procedure: 1) Compute f′(x). 2) Find critical points by solving f′(x) = 0 or where f′ is undefined. 3) Make a sign chart (test points between critical values and check endpoints if the domain’s closed). 4) Record intervals where f′ is positive (increasing) or negative (decreasing). Use the first-derivative test at each critical point to classify local extrema and remember nondifferentiable points (cusps) can change behavior. This is exactly what the AP CED expects for FUN-4.A (justify using derivative sign and tests). For a quick study guide, see Topic 5.3 (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y). For more review and practice problems, check the Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5) and thousands of practice questions (https://library.fiveable.me/practice/ap-calculus).
When do I use the first derivative test vs just looking at the graph?
Use the graph when it’s given and clear: if you have f(x) plotted on the exam, you can read where it rises/falls directly (look for left-to-right up = increasing, down = decreasing), and use visible endpoints and cusps. Use the first-derivative test when you only have a formula, need justification, or when the graph is ambiguous (flat spots, cusps, vertical tangents, nondifferentiable points). Procedure: find critical points (where f′ = 0 or undefined), make a sign chart for f′ on intervals between them (or test points), then apply the first derivative test to classify local max/min (f′ changes +→− for a local max, −→+ for a local min). Remember to check endpoints for absolute extrema. This matches CED keywords: critical point, derivative sign, test point, first derivative test, differentiability. For a quick review, see the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and try practice problems (https://library.fiveable.me/practice/ap-calculus) to get fluent at sign charts and AP-style justifications.
What's the difference between finding critical points and finding increasing/decreasing intervals?
Good question—they’re related but not the same job. - Critical points are the x-values where f′(x) = 0 or f′ is undefined (and where f is in the domain). They’re just candidates—stationary points (f′ = 0) and nondifferentiable points. Finding them is Step 1. (See Topic 5.2 for more on critical points.) - Determining increasing/decreasing intervals uses those critical points plus sign analysis of f′. You make a sign chart (or test points) on each interval between critical points (and include endpoints on closed intervals) to see where f′>0 (f increasing) or f′<0 (f decreasing). That gives the actual intervals of increase/decrease and lets you apply the First Derivative Test for local extrema (Topic 5.4 / FUN-4.A). AP tip: always justify using derivative sign and mention nondifferentiable endpoints if they affect behavior—that’s what the CED expects (FUN-4.A.1). For a quick review, check the Topic 5.3 study guide (Fiveable) here: (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y). For extra practice, try problems at (https://library.fiveable.me/practice/ap-calculus).
Can someone explain step by step how to find intervals where f(x) is increasing?
Step-by-step: how to find where f(x) is increasing 1. Compute f ′(x). The first derivative gives the function’s rate of change (FUN-4.A). 2. Find the domain of f and f ′. Note points where f′(x)=0 or f′ is undefined—those are critical points. 3. Put the critical points (and any endpoints if you have a closed interval) in order and make intervals between them. 4. Pick one test x-value in each interval and evaluate the sign of f′(test). If f′>0 on an interval, f is increasing there; if f′<0, f is decreasing. (This is the sign-chart / first derivative test idea.) 5. State the final answer as a union of open intervals where f′>0; include endpoints only if the interval is closed and the derivative/sign supports it. 6. Mention nondifferentiable cusps or jumps as special cases (check the function itself). This is exactly what AP expects: use the derivative’s sign to justify increasing/decreasing intervals (see Topic 5.3 study guide for examples) (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y). For more review across Unit 5, see (https://library.fiveable.me/ap-calculus/unit-5) and try practice problems (https://library.fiveable.me/practice/ap-calculus).
I don't understand why f'(x) > 0 means the function is increasing - can someone explain this?
Think of f′(x) as the instantaneous rate of change (slope) of f. If f′(x) > 0 on an interval, every tangent line there has positive slope, so small moves to the right always increase f. More formally, if f is differentiable on (a,b) and f′(x) > 0 for all x in (a,b), the Mean Value Theorem says for any x1
How do I solve for where f'(x) > 0 and f'(x) < 0?
Solve f′(x) > 0 or f′(x) < 0 in 4 clear steps: 1. Compute f′(x). Use derivative rules (power, product, chain, etc.). 2. Find critical points: solve f′(x) = 0 and note where f′ is undefined. These split the domain. (Include endpoints of the domain too.) 3. Make a sign chart: pick one test x in each interval between critical points and evaluate f′(test). If f′(test) > 0 that whole interval is increasing; if f′(test) < 0 it’s decreasing. Don’t forget intervals where f′ is undefined—they can change sign. 4. Use the First Derivative Test to classify local extrema: sign change +→− gives a local max, −→+ gives a local min. State interval endpoints separately if the interval is closed. On the AP exam, justify your sign choices (show test points) and include nondifferentiable points as candidates (CED FUN-4.A). For a refresher and examples, see the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and more practice at (https://library.fiveable.me/practice/ap-calculus).
What do I do after I find the critical points to determine increasing/decreasing intervals?
After you find the critical points (where f′ = 0 or f′ is undefined), make a sign chart for f′ to determine increasing/decreasing intervals: 1. List the critical points and any domain/endpoints in order on the number line. 2. Pick one test point in each open interval between them. 3. Evaluate f′ at each test point: - If f′(test) > 0 on an interval, f is increasing there. - If f′(test) < 0, f is decreasing there. 4. Use the First Derivative Test to classify each critical point: - f′ changes + → −: local max. - f′ changes − → +: local min. - No sign change: no local extremum (could be inflection or flat spot). 5. On closed intervals, also check endpoints (candidates for absolute extrema). 6. If f′ is undefined at a point, consider nondifferentiability (cusp, corner) before concluding. This is exactly the FUN-4.A/CED approach—you can review an example and practice problems in the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and try more problems at (https://library.fiveable.me/practice/ap-calculus).
Why do I need to test points in each interval instead of just finding where f'(x) = 0?
Good question—setting f′(x)=0 only finds critical (stationary) points; it doesn’t tell you whether f actually goes up or down on either side. A sign chart (pick one test point in each interval between critical points and non-differentiable points) tells you whether f′>0 (increasing) or f′<0 (decreasing) on that whole interval. That matters because at some critical points f′ changes sign (local max/min) and at others it doesn’t (flat inflection). Also include points where f′ is undefined and endpoints as interval boundaries. On the AP exam you’re expected to justify interval behavior using derivative signs (FUN-4.A, first derivative test), so show the sign work or a quick test-value calculation. For a focused review, see the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and try practice problems (https://library.fiveable.me/practice/ap-calculus).
How do I write interval notation for where a function is increasing or decreasing?
Find where f′(x) > 0 (increasing) and where f′(x) < 0 (decreasing). Steps and how to write them in interval notation: 1. Find domain, then solve f′(x) = 0 and where f′ is undefined → list critical points. 2. Make a sign chart: split the domain at the critical points and pick a test point in each subinterval to evaluate sign of f′. 3. Record intervals where f′>0 (increasing) and where f′<0 (decreasing). Use parentheses for open intervals between critical points: e.g., (−∞, a), (a, b), (b, ∞). If an endpoint is in the domain and the derivative sign on the adjacent open interval shows increasing/decreasing up to that endpoint, include the endpoint with a bracket: [a, b) or (a, b] as appropriate. (You only use a bracket if the endpoint is part of the function’s domain and you want to include it.) 4. State your result: “f is increasing on (−∞, −1) ∪ (2, 5] and decreasing on (−1, 2).” Always justify by citing sign of f′ on each piece (first-derivative test / sign chart). This aligns with FUN-4.A in the CED. For a worked example and practice, see the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and try problems from the Unit 5 page (https://library.fiveable.me/ap-calculus/unit-5) or the practice bank (https://library.fiveable.me/practice/ap-calculus).
I'm confused about sign charts - how do I make one for the first derivative?
Make a sign chart for f′ in four simple steps: 1. Find critical numbers: solve f′(x)=0 and note where f′ is undefined. These partition the domain into open intervals (include endpoints if you have a closed interval). (CED keywords: critical point, differentiability, cusp.) 2. List the intervals on a number line and pick one test point in each interval (not the endpoints). (CED keyword: test point.) 3. Evaluate f′ at each test point (or determine sign from a graph/table). Put + if f′>0 (f increasing) or − if f′<0 (f decreasing) on that whole interval. If f′ changes sign at a critical number, use the First Derivative Test to classify local max/min; if it doesn’t change sign, no local extremum. (CED: first derivative test, local maximum, local minimum.) 4. Don’t forget endpoints and places where f isn’t differentiable—they can be candidates for absolute extrema. Also watch for cusps/vertical tangents where f′ is undefined. On the AP exam you may get algebraic, graphical, or tabular f′ information, so be ready to use arithmetic or reasoning. For extra practice and worked examples, see the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and more practice problems (https://library.fiveable.me/practice/ap-calculus).
What happens at critical points - are they included in increasing or decreasing intervals?
Short answer: critical points themselves aren’t counted as points “where a function is increasing” or “decreasing”—you describe increasing/decreasing on intervals, and those intervals are usually open (e.g., (a, c) or (c, b)). A critical point is where f′(x)=0 or f′ is undefined; use it as a boundary when making a sign chart for f′. How to report on the exam (CED/FUN-4.A language): - Find critical points (f′=0 or f′ undefined). - Test sign of f′ on intervals between those critical points (pick test points). If f′>0 on (p,q) then f is increasing on (p,q); if f′<0 then f is decreasing on (p,q). - Don’t include the critical point in the open interval answer. If the problem asks for behavior on a closed domain or wants endpoints, you may state inclusion of endpoints separately (e.g., increasing on [a,c) if a is the domain endpoint). - Watch for non-differentiable features (cusps, corners)—they’re critical points too and must be checked. For practice and a quick refresher, see the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and try problems at (https://library.fiveable.me/practice/ap-calculus).
How do I find increasing and decreasing intervals if the derivative has fractions or radicals?
Short answer: treat fractional or radical derivatives the same way—find where f′ = 0 or undefined, break the domain into intervals, pick test points, and use a sign chart. How-to steps (quick): - First get the algebra under control: simplify f′ as much as possible (factor numerator, combine to a single fraction, and simplify radicals). - Find critical numbers: solve numerator = 0 and note x where denominator = 0 or a radical’s inside becomes invalid (these make f′ undefined). Also keep the original function’s domain in mind. - Make a sign chart: list all critical points and domain endpoints in order. For each open interval, pick an easy test point and evaluate the sign of f′ (you can often just check signs of factors rather than full values). If f′>0 the function’s increasing there; if f′<0 it’s decreasing. - Watch endpoints, points where f′ is undefined (possible cusps/vertical tangents), and use the first-derivative test to classify local extrema (FUN-4.A in the CED). This is exactly what AP expects for Topic 5.3—show the sign reasoning/justification (first-derivative test, critical points). For extra worked examples and practice, see the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and more Unit 5 resources (https://library.fiveable.me/ap-calculus/unit-5). For lots of practice problems, try (https://library.fiveable.me/practice/ap-calculus).
Can I use my calculator to find where f'(x) is positive or negative?
Yes—your graphing calculator is a helpful tool, but it’s not a substitute for the math you must justify on the exam. How a calculator helps - Graph f and/or f′ and use the TRACE/ROOT features to find zeros of f′ (critical points) and sample values to see where f′>0 or f′<0. - Use numerical derivative routines or difference quotients to approximate f′(x) at specific x (good for nonsymbolic functions). - Make a sign chart quickly by evaluating f′ (or slopes of f) at test points between critical values. Important AP/CEC notes (CED-aligned) - You still need to justify conclusions about increasing/decreasing using derivative sign, test points, and first-derivative test (FUN-4.A; keywords: derivative sign, critical point, test point). - On the AP exam, calculators are allowed only on certain parts (multiple-choice Part B and some free-response Part A). You must be ready to do analyses without a calculator for non-calculator sections. - Watch for nondifferentiable points (cusps, corners) and endpoints as candidates when deciding intervals. Practice: review Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and try problems (https://library.fiveable.me/practice/ap-calculus) so you can both use the calculator effectively and justify answers without it.
I keep getting the wrong intervals - what are the most common mistakes when finding where functions increase or decrease?
Common mistakes and how to avoid them: - Forgetting the domain/endpoints. Always restrict critical points and sign charts to where f is defined (include endpoints on closed intervals). AP problems expect you to justify behavior on the given domain (CED FUN-4.A). - Missing critical points where f′ is undefined. Points where f′ = 0 or f′ DNE can change sign—include both. - Bad algebra when solving f′(x)=0. Re-check factoring, canceling, and extraneous solutions; plug suspects back into f′. - Skipping the sign chart or using too few test points. Make a sign chart with intervals determined by every critical point and choose one test value per interval to check f′ sign. - Using f instead of f′ (or vice versa) to decide increase/decrease. Remember: f′>0 ⇒ f increasing; f′<0 ⇒ f decreasing (FUN-4.A.1). - Ignoring nondifferentiable cusps/vertical tangents. If f′ DNE, test either side for sign change. - Not reporting intervals cleanly (open vs closed) or not justifying conclusions. On the exam, state your intervals and give brief reasons (first derivative test or sign chart). If you want a step-by-step refresher, check the Topic 5.3 study guide (https://library.fiveable.me/ap-calculus/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y) and drill problems at https://library.fiveable.me/practice/ap-calculus.