In previous guides, we learned all about making conclusions regarding the behavior of a function based on the behavior of its derivatives such as whether the function is increasing or decreasing at a point, concave up or concave at a point, and more! While we mostly focused on algebraically determining the behavior of functions, we can also determine information graphically! The key features of the graphs of , , and are all related to one another. 🔑
Let’s dive into how we can do that!
Given the graphs of , , and or some combination of the three, we can determine information about another much as we did so algebraically. The knowledge you learned in our previous Unit 5 subtopic guides can be carried over to this subtopic—instead of using the equations for , , and , you can look at (one of) their graphs and see where the x-axis is crossed or where the graph is positive or negative, increasing or decreasing, etc, to infer information about the other graphs.
📌 Here is a list of specific guides to revisit:
- 5.3 Determining Intervals on Which a Function Is Increasing or Decreasing
- 5.4 Using the First Derivative Test to Determine Relative (Local) Extrema
- 5.6 Determining Concavity of Functions over Their Domains
- 5.7 Using the Second Derivative Test to Determine Extrema
📉 Trends and Concavity
Here’s a quick summary of what you’ve learned so far in this unit about trends and concavity:
- When a function is increasing, the first derivative will be positive ().
- When a function is decreasing, the first derivative will be negative ().
- When a function is concave up, the second derivative will be positive () and the first derivative will increase.
- When a function is concave down, the second derivative will be negative () and the first derivative will decrease.
👀 Trends and Concavity Graphically
Let’s apply this information to the following graph of a function, .
Taking a look at this graph, we can describe between each interval:
- From and , is decreasing, so is negative.
- From and , is increasing, so is positive.
What about ? Let’s take a look at the concavity of :
- From and , is concave up. Therefore, is positive and is increasing.
- From , is concave down. Therefore, is negative and is decreasing.
🔍 Extrema and Points of Inflection
Based on where the graph of the function changes direction and concavity, we can also interpret maximums, minimums, x-intercepts, and points of inflection of the graphs of the first and second derivatives.
- If has a relative minimum (the graph changes from decreasing to increasing), then will change from negative to positive at that point.
- If has a relative maximum (the graph changes from increasing to decreasing), then will change from positive to negative at that point.
- If has a point of inflection, changing from concave up to concave down, then will have a relative maximum and will change from positive to negative at that point.
- If has a point of inflection, changing from concave down to concave up, then will have a relative minimum and will change from negative to positive at that point.
If we boil this down to two key concepts, realize that:
- All relative extrema of are x-intercepts of .
- All points of inflection of are relative extrema of .
This may seem like a lot, but once you see it in action, it’ll make more sense! ⬇️
👀 Extrema and POIs Graphically 1
Here’s a relatively easy example! The derivative of the differentiable function , , is graphed.
What can we tell about at the point based on the graph of its derivative ?
By looking at the graph of , we see that crosses the x-axis at the point of interest . It is negative before and positive after . This means that is decreasing before the point and increasing after it, indicating that the point is a relative minimum on the graph of .
The justification we used above to determine the answer is essentially just applying the First Derivative Test but in graphical form! Here’s a quick look at and so you can really see their relationship:
👀 Extrema and POIs Graphically 2
Now, let’s take another look at the example before, but focus on relative extrema and points of inflection.
You’ll notice the following:
- At and , has relative minima. Therefore, has x-intercepts at these points and will change from negative to positive.
- At , has a relative maximum. has another x-intercept, but the opposite is true: will change from positive to negative.
- At , has a point of inflection, changing from concave up to concave down. This means will have a relative maximum and has an x-intercept changing from positive to negative at .
- At , has an inflection point, but it changes from concave down to concave up. Therefore, has a relative minimum and has an x-intercept changing from negative to positive at .
👀 Extrema and POIs Graphically 3
Before you move on to taking a look at graphs yourself, take a look at the following graph of and think about:
- What happens to at , denoted by the red dotted line?
- What happens to at , denoted by the black dotted line?
- What happens to at ?
At , has a relative maximum. This tells us that will have an x-intercept at this point and change from positive to negative!
At , changes from being concave down to concave up. This tells us that will have a relative minimum at this point and has an x-intercept, changing from negative to positive.
Take a look at in blue and in green! You can see exactly these trends.
And here’s a graph with added as well, denoted in purple.
📝Practice Problems
Now it’s time for you to do some practice on your own! These won’t be as tough, they will more generally test your knowledge of these trends.
❓Practice Problems
Question 1:
The second derivative of the differentiable function , , is graphed.
Given that , what can we tell about at the point based on the graph of its second derivative ?
Question 2:
The second derivative of the differentiable function , , is graphed.
What can we tell about at the point based on the graph of its second derivative ?
✅ Answers and Solutions
Question 1:
Answer: has a relative minimum at the point .
Solution:
By looking at the graph of , we see that is positive at the point of interest . This means that is concave up at the point. Combined with the fact that , we can apply the Second Derivative Test to conclude that has a minimum at .
Question 2:
Answer: is concave down at the point .
Solution:
By looking at the graph of , we infer that is negative at the point of interest . This means that is concave down at the point.
⭐ Closing
Woah! You made it to the end of this guide. To practice with some of this material, we recommend getting into Desmos and graphing a function, its first derivative, and its second derivative to see the features of each. Good luck! 🍀
Vocabulary
The following words are mentioned explicitly in the College Board Course and Exam Description for this topic.
| Term | Definition |
|---|---|
| derivative | The instantaneous rate of change of a function at a specific point, representing the slope of the tangent line to the function at that point. |
| function behavior | The characteristics of a function including its increasing/decreasing intervals, concavity, extrema, and end behavior. |
| key features | Important characteristics of a function including extrema, inflection points, intervals of increase/decrease, and concavity. |
Frequently Asked Questions
How do I know if a function is increasing or decreasing using the first derivative?
Use f′ to check slope: if f′(x) > 0 on an interval, f is increasing there (tangent slopes are positive); if f′(x) < 0, f is decreasing (tangent slopes are negative). Practical steps you’ll use on the AP: 1. Find critical numbers: solve f′(x)=0 and where f′ is undefined (these are candidates for sign changes). 2. Make a sign chart for f′ (pick test points in each interval between critical numbers). 3. Read off behavior: + means increasing, − means decreasing. Use the First Derivative Test to classify local max/min when f′ changes sign at a critical point. 4. Remember nondifferentiable points can still be local extrema—include them as candidates. Also note: constant f′=0 on an interval means f is constant there. This procedure is exactly what Topic 5.9/FUN-4.A expects—justify behavior from derivatives on the exam. For a quick review, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te). Want practice? Try lots of sign-chart problems on Fiveable’s unit page (https://library.fiveable.me/ap-calculus/unit-5) or the practice bank (https://library.fiveable.me/practice/ap-calculus).
What's the difference between f, f prime, and f double prime on a graph?
Think of the three graphs as different layers of information about the same curve. - f(x): the actual height. Peaks/troughs are visible here (local max/min) and horizontal tangents occur where the slope is 0 (stationary points). - f′(x): gives the slope of f. Where f′>0, f is increasing; where f′<0, f is decreasing. Zeros of f′ are candidates for local extrema—use sign changes in f′ (first derivative test) to decide. - f′′(x): tells about curvature (concavity). If f′′>0, f is concave up (slope of f is increasing); if f′′<0, f is concave down (slope of f is decreasing). Points where f′′ changes sign are inflection points. Connections to use on the AP: draw sign charts for f′ and f′′ to identify increasing/decreasing intervals, local extrema, and concavity (CED keywords: critical point, inflection point, first/second derivative test). For targeted practice and worked examples see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and try problems at (https://library.fiveable.me/practice/ap-calculus).
When do I use the second derivative test vs the first derivative test?
Use the second-derivative test when you have a critical number c where f′(c)=0 and f″ is defined at c. If f″(c)>0, f has a local minimum at c; if f″(c)<0, f has a local maximum. If f″(c)=0 (or f″ doesn’t exist), the second-derivative test is inconclusive. Use the first-derivative test when f′ changes sign around a critical point or when f′ or f″ aren’t nicely behaved. Make a sign chart for f′(x): if f′ goes + → − at c, you have a local max; − → + gives a local min; no sign change means no local extremum. The first-derivative test also handles stationary points where f′ is undefined or when f″(c)=0. On the AP exam, you’ll need to justify your choice (CED FUN-4.A, FUN-4.A.11). For review and examples, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5). For more practice, check the AP problems pool (https://library.fiveable.me/practice/ap-calculus).
How can I tell where a function has a maximum or minimum just by looking at f'(x)?
Find the critical points first: where f′(x)=0 or f′ is undefined. Then make a sign chart for f′ (or examine its graph). - If f′ changes from positive to negative at a critical point (f′: + → −), f is increasing before and decreasing after → local maximum. - If f′ changes from negative to positive (f′: − → +), f has a local minimum. - If f′ doesn’t change sign (stays + or stays −), the point is not a local extremum (a stationary point). You can also use the second-derivative test when f″ exists: at a critical point x0 with f′(x0)=0, if f″(x0)>0 → local min (concave up); if f″(x0)<0 → local max (concave down). Remember to check endpoints and points of nondifferentiability for absolute extrema (Candidates Test). For AP-style practice and a short CED-aligned review of Topic 5.9, see the Fiveable study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and extra practice problems (https://library.fiveable.me/practice/ap-calculus).
I'm confused about concavity - how does the second derivative show if a graph curves up or down?
Think of concavity as which way the graph "holds water." If f''(x) > 0 on an interval, f is concave up there (like a cup)—slopes (f') are increasing. If f''(x) < 0, f is concave down (like an upside-down cup)—slopes are decreasing. An inflection point is where f'' changes sign (so the curve switches from curving up to down or vice versa). How to use it on the AP: check f''(x) or make a sign chart for f'' to identify concave-up / concave-down intervals and candidate inflection points (FUN-4.A, CED keywords: concavity, inflection point, sign chart). Also remember the second-derivative test: if f'(c)=0 and f''(c)>0, local min; if f''(c)<0, local max. For more examples and AP-style practice, review Topic 5.9 (study guide) (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and try problems at (https://library.fiveable.me/practice/ap-calculus).
What does it mean when f'(x) = 0 and how do I find those points?
When f′(x) = 0 at some x = c, the tangent to f at c is horizontal—c is a stationary (critical) point and a candidate for a local max, local min, or neither (a plateau). To find these points: 1. Compute f′(x). 2. Solve f′(x) = 0 for x (and also list x where f′ is undefined—those are critical numbers too). 3. Put those x-values on a number line and make a sign chart for f′(x) to see where f is increasing (f′>0) or decreasing (f′<0). The First Derivative Test: if f′ changes +→− at c, f has a local max; −→+ gives a local min; no sign change → no extremum. 4. Optionally use the Second Derivative Test: if f″(c) > 0 then local min, if f″(c) < 0 then local max; if f″(c) = 0 the test is inconclusive (use sign chart). 5. Remember to check endpoints of closed intervals and points of nondifferentiability. This is exactly what the CED calls critical numbers, stationary points, and the first/second derivative tests (Topic 5.9). For extra worked examples and practice, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5). For lots of practice problems, try Fiveable’s practice page (https://library.fiveable.me/practice/ap-calculus).
How do I connect the graph of a function to its derivative graph step by step?
Step-by-step: start with slopes, then signs, then concavity. 1. Look at f’s tangent slopes: where f’s slope is positive, f′ is above the x-axis; where slopes are negative, f′ is below. Horizontal tangents on f (slope 0) = x-intercepts of f′ (critical points). 2. Classify critical points: if f′ changes + → − at a zero, f has a local max; − → + gives a local min (first-derivative test). If f′ = 0 and f″(x) > 0, it's a local min; if f″(x) < 0, local max (second-derivative test). 3. Concavity: where f′ is increasing (f′′ > 0), f is concave up; where f′ is decreasing (f′′ < 0), f is concave down. Inflection points are where f′ changes direction (sign of f′′ changes) and f is continuous. 4. Make sign charts: mark f′ sign to get increasing/decreasing intervals and f′′ sign to get concavity intervals. Check points of nondifferentiability separately. For AP practice and examples that follow CED keywords (critical point, inflection, first/second derivative tests), see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5). For extra problems, use the practice page (https://library.fiveable.me/practice/ap-calculus).
Can someone explain why inflection points happen when f''(x) = 0?
An inflection point is where the graph of f changes concavity (concave up ⇄ concave down). f''(x) measures concavity: f''(x) > 0 means concave up, f''(x) < 0 means concave down. So an inflection point must occur at a point where f'' either is 0 or doesn't exist, because those are the only places the sign of f'' can switch. Crucially, f''(x)=0 alone is not enough—you must check that f'' actually changes sign at that x. If f''(x)=0 but stays positive (or stays negative) on both sides, there’s no inflection point. On the AP exam, justify concavity changes using sign charts for f'' (CED keywords: concavity, inflection point, sign chart for f''). For extra practice and examples (including sign-chart strategies), see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and try practice problems (https://library.fiveable.me/practice/ap-calculus).
What's the relationship between where f(x) has a max/min and where f'(x) crosses the x-axis?
If f has a local max or min at x = c, then c is usually a critical number: f′(c) = 0 or f′ is undefined. How f′ crosses the x-axis tells you which: - If f′ changes from positive to negative at c (f′>0 before, f′<0 after), f rises then falls → local maximum. - If f′ changes from negative to positive at c (f′<0 before, f′>0 after), f falls then rises → local minimum. - If f′ = 0 but does NOT change sign (e.g., f′ touches 0 and stays positive), there’s no local extremum—often a horizontal inflection or higher-order stationary point. You can also use the second derivative test: if f′(c)=0 and f″(c)>0 → local min; if f″(c)<0 → local max; if f″(c)=0 the test is inconclusive (use the first-derivative sign chart). These are exactly the skills AP asks you to justify (first/second derivative tests, sign charts, concavity). For a quick refresher, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and try practice problems (https://library.fiveable.me/practice/ap-calculus).
I don't understand how to read derivative graphs - how do I know what the original function looks like?
Think of f′(x) as the slope of f and f″(x) as how that slope is changing. - If f′ > 0 on an interval, f is increasing there; if f′ < 0, f is decreasing. Use a sign chart for f′ to mark increasing/decreasing intervals (critical numbers occur where f′ = 0 or undefined). - When f′ crosses from + to − at a critical number you get a local max; − to + gives a local min (first-derivative test). - The slope of the f′ graph = f″. If f′ is increasing (f′ slope > 0, so f″ > 0) then f is concave up; if f′ is decreasing (f″ < 0) f is concave down. - Inflection points occur where concavity changes (often where f′ has a local extremum or where f″ = 0/undefined). - Watch for nondifferentiable behavior: corners, cusps, vertical tangents on f correspond to breaks or vertical behavior in f′. These are exactly the CED links between f, f′, f″ used on the exam (FUN-4.A, FUN-4.A.11). For worked examples and practice, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te), the Unit 5 overview (https://library.fiveable.me/ap-calculus/unit-5), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).
When f'(x) is positive vs negative, what does that tell me about the original function?
If f′(x) > 0 on an interval, f is increasing there—its tangent slopes are positive, so as x grows f(x) rises. If f′(x) < 0, f is decreasing—tangents slope downward and f(x) falls. Points where f′(x) = 0 or f′ is undefined are critical numbers and potential local extrema; use the First Derivative Test (check sign of f′ before and after) to decide if you have a local max or min (CED keywords: critical point, first derivative test). Also remember f′ tells slope while f″ tells how that slope changes (concavity): f″ > 0 → f′ increasing → f concave up; f″ < 0 → f′ decreasing → f concave down. These ideas (FUN-4.A) show up in Unit 5 questions on the AP exam—practice them with sign charts and sketching. For a focused review see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te), unit overview (https://library.fiveable.me/ap-calculus/unit-5), and lots of practice problems (https://library.fiveable.me/practice/ap-calculus).
How do I use the second derivative to determine if a critical point is a max or min?
If c is a critical number (usually f'(c) = 0) and f is twice-differentiable near c, use the second-derivative test: - If f''(c) > 0, the graph is concave up at c → local minimum at c. (Think: tangent sits below the curve.) - If f''(c) < 0, the graph is concave down at c → local maximum at c. (Tangent sits above the curve.) - If f''(c) = 0 (or f'' doesn't exist), the test is inconclusive—use the first-derivative test or check higher derivatives/graph behavior. Example: if f'(c)=0 and f''(c)=4 (>0), then f has a local min at c. Remember: the second-derivative test requires f'' to exist at c; critical points can also occur where f' is undefined, and then you can’t apply this test. This matches CED keywords (critical point, concavity, second derivative test, first derivative test). For more review and examples, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and try practice problems (https://library.fiveable.me/practice/ap-calculus).
What happens to the graph of f(x) when f''(x) changes from positive to negative?
If f''(x) changes from positive to negative at a point a, the graph of f changes concavity there: it goes from concave up (f''>0) to concave down (f''<0). That change in concavity means x = a is an inflection point—but remember: f''(a)=0 alone isn’t enough; you need the sign of f'' to actually change. Interpretations you should know from the CED (FUN-4.A): - While f''>0 on the left, f' is increasing; while f''<0 on the right, f' is decreasing. So f' has a local maximum at a (if f'' is continuous). - If additionally f'(a)=0, then f has a local maximum at a (second derivative test: f''(a)<0 indicates a local max). For AP-style answers, justify concavity and the inflection using sign charts for f' and f'' (see the Topic 5.9 study guide on Fiveable for examples) (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te). For extra practice, try problems on Unit 5 (https://library.fiveable.me/ap-calculus/unit-5) or the large problem set (https://library.fiveable.me/practice/ap-calculus).
Why does my teacher keep asking about the behavior of f, f', and f'' together on the same problem?
Because f, f′, and f″ together tell the full story of a graph. f′ shows where f is increasing/decreasing (sign chart → critical points); f″ shows concavity and inflection points. Combining them lets you: - locate and classify extrema (first-derivative test or second-derivative test when f′(c)=0 and f″(c)≠0), - decide whether a stationary point is a max, min, or inflection, - sketch accurate shapes (slope from f′, curvature from f″), - and justify conclusions on the AP (CED learning objective FUN-4.A). Your teacher asks about all three because AP free-response and multiple-choice expect you to connect these representations and give reasoning (e.g., “f′ changes from + to − at c, so f has a local max; f″<0 near c so f is concave down”). Practice making sign charts for f′ and f″, mark critical numbers and points of nondifferentiability, and state units/meaning when context appears. For a focused review, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and try practice problems (https://library.fiveable.me/practice/ap-calculus).
How do I sketch f'(x) if I'm only given the graph of f(x)?
Start by reading f’s graph like a sign/slope map for f′: - Where f is increasing, draw f′ above the x-axis (positive); where f is decreasing, draw f′ below (negative). Use longer/steeper tangents on f to make larger magnitude for f′, shallower tangents → values of f′ near zero. - Mark zeros of f′ at f’s critical points: any place f has a horizontal tangent or a sharp top/bottom (local max/min or cusp) → put f′ = 0 (or undefined at a cusp). - Inflection points of f (where concavity changes) correspond to zeros or slope changes of f′: at an inflection, f′ has a local extremum (its slope changes sign). So if f changes from concave up to concave down, draw f′ crossing from increasing to decreasing (a local max of f′). - Check nondifferentiable points on f—f′ will be undefined there. - Finally, sketch f′ smoothly, matching signs, zeros, and relative sizes; don’t invent oscillation where f is smooth. These are exactly the CED connections (increasing/decreasing, critical points, inflection/concavity, first/second derivative tests). For examples and practice, see the Topic 5.9 study guide (https://library.fiveable.me/ap-calculus/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te) and more practice problems (https://library.fiveable.me/practice/ap-calculus).