---
title: "AP Calculus 5.9: Connecting f, f', and f''"
description: "Review AP Calculus 5.9 and how f, f', and f'' connect, including increasing and decreasing behavior, concavity, extrema, inflection points, and graph-based justifications."
canonical: "https://fiveable.me/ap-calc/unit-5/connecting-a-function-its-first-derivative-and-its-second-derivative/study-guide/NQXfonM48ssVKKRHQ7Te"
type: "study-guide"
subject: "AP Calculus AB/BC"
unit: "Unit 5 – Analytical Applications of Differentiation"
lastUpdated: "2026-06-09"
---

# AP Calculus 5.9: Connecting f, f', and f''

## Summary

Review AP Calculus 5.9 and how f, f', and f'' connect, including increasing and decreasing behavior, concavity, extrema, inflection points, and graph-based justifications.

## Guide

The graphs of a [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") $f$, its [first derivative](/ap-calc/key-terms/first-derivative "fv-autolink") $f'$, and its second derivative $f''$ are all linked. The sign of $f'$ tells you where $f$ is increasing or decreasing, the sign of $f''$ tells you concavity, and the places where $f'$ or $f''$ cross the x-axis point to extrema and inflection points. For AP Calculus, name which graph you are using before making a conclusion about $f$.

## Why This Matters for the AP Calculus Exam

This topic pulls together everything from earlier in [Unit 5](/ap-calc/unit-5 "fv-autolink") and asks you to reason graphically instead of only algebraically. On the AP Calculus exam, you often get the graph of $f'$ (or sometimes $f''$) and have to draw conclusions about $f$ without ever seeing a formula. That shows up in both multiple-choice questions and free-response questions, where you may need to justify where $f$ has a maximum, a minimum, or a point of inflection.

The justifications you write here matter for clear exam work. Reasoning like "$f$ is [concave up](/ap-calc/key-terms/concave-up "fv-autolink") on this [interval](/ap-calc/unit-10-infinite-sequences-and-series-bc-only-/finding-taylor-polynomial-approximations-functions/study-guide/LszguYzKz0M6GdqTRSr6 "fv-autolink") because $f'$ is increasing there" is exactly the kind of language graders look for. Refer to $f$, $f'$, and $f''$ by name, not "it" or "the function," so your argument is easy to follow.

## Key Takeaways

- Where $f'$ is positive, $f$ is increasing; where $f'$ is negative, $f$ is decreasing.
- Where $f''$ is positive, $f$ is concave up and $f'$ is increasing; where $f''$ is negative, $f$ is [concave down](/ap-calc/key-terms/concave-down "fv-autolink") and $f'$ is decreasing.
- [Relative extrema](/ap-calc/key-terms/relative-extrema "fv-autolink") of $f$ happen where $f'$ crosses the x-axis (changes sign).
- [Points of inflection](/ap-calc/key-terms/points-of-inflection "fv-autolink") of $f$ line up with relative extrema of $f'$ and with sign changes of $f''$.
- A [critical point](/ap-calc/key-terms/critical-point "fv-autolink") occurs where $f'=0$ or $f'$ does not exist (such as a cusp or vertical [tangent](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt "fv-autolink")).
- The First and Second Derivative Tests both work in graphical form, not just algebraic form.

## Connecting the Three Graphs

Given the graphs of $f$, $f'$, and $f''$, or just one of them, you can read off information about the others. Instead of using equations, you look at where a graph crosses the x-axis, where it is positive or negative, and where it is increasing or decreasing.

Guides worth revisiting before this one:

- [5.3 Determining Intervals on Which a Function Is Increasing or Decreasing](/ap-calc/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y)
- [5.4 Using the First Derivative Test to Determine Relative (Local) Extrema](/ap-calc/unit-5/using-first-derivative-test-to-determine-relative-local-extrema/study-guide/BjnQNCShz0uQiZGhSl2g)
- [5.6 Determining Concavity of Functions over Their Domains](/ap-calc/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V)
- [5.7 Using the Second Derivative Test to Determine Extrema](/ap-calc/unit-5/using-second-derivative-test-determine-extrema/study-guide/g8rSveVmLeEuKLK4T7ep)

### Trends and Concavity

Here is a quick summary of how the three graphs connect:

- When $f$ is increasing, $f'$ is positive ($>0$).
- When $f$ is decreasing, $f'$ is negative ($<0$).
- When $f$ is concave up, $f''$ is positive ($>0$) and $f'$ is increasing.
- When $f$ is concave down, $f''$ is negative ($<0$) and $f'$ is decreasing.

Apply this to the graph of a function $g(x)$ below.

![Graph of g(x)](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled.png-1704919445168-25212)

###### Graph of function $g(x)$, courtesy of Zweig Media

Describing $g'(x)$ across intervals:

- On $(-\infty,-2)$ and $(0.85,2.8)$, $g(x)$ is decreasing, so $g'(x)$ is negative.
- On $(-2, 0.85)$ and $(2.8,\infty)$, $g(x)$ is increasing, so $g'(x)$ is positive.

Now describing $g''(x)$ using concavity:

- On $(-\infty,-0.5)$ and $(1.5, \infty)$, $g(x)$ is concave up, so $g''(x)$ is positive and $g'(x)$ is increasing.
- On $(-0.5,1.5)$, $g(x)$ is concave down, so $g''(x)$ is negative and $g'(x)$ is decreasing.

### Extrema and Points of Inflection

Where the graph of $f$ changes direction or concavity, you can pin down maxima, minima, x-intercepts, and inflection points on the graphs of $f'$ and $f''$.

- If $f(x)$ has a [relative minimum](/ap-calc/key-terms/relative-minimum "fv-autolink") (changes from decreasing to increasing), then $f'(x)$ changes from negative to positive there.
- If $f(x)$ has a [relative maximum](/ap-calc/key-terms/relative-maximum "fv-autolink") (changes from increasing to decreasing), then $f'(x)$ changes from positive to negative there.
- If $f(x)$ has a point of inflection going from concave up to concave down, then $f'(x)$ has a relative maximum and $f''(x)$ changes from positive to negative there.
- If $f(x)$ has a point of inflection going from concave down to concave up, then $f'(x)$ has a relative minimum and $f''(x)$ changes from negative to positive there.

Two big ideas to remember:

1. All relative extrema of $f(x)$ are x-intercepts of $f'(x)$.
2. All points of inflection of $f(x)$ are relative extrema of $f'(x)$.

## How to Use This on the AP Calculus Exam

### Problem Solving

When you are handed a graph of $f'$, read it like a sign chart. Find where it crosses the x-axis, then check whether it goes from negative to positive (relative minimum of $f$) or positive to negative (relative maximum of $f$). This is the First Derivative Test in graphical form.

When you are handed a graph of $f''$, check its sign to find concavity, and find where it crosses the x-axis (and changes sign) to locate inflection points of $f$.

### Worked Example: Reading a Graph of $f'$

The derivative $f'$ of the [differentiable function](/ap-calc/key-terms/differentiable-function "fv-autolink") $f$ is graphed below.

![Graph of f prime](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled_1.png-1704919445177-10915)

###### Graph of the derivative of $f$ with the point $(1.5, 0)$ labeled. Image created with Desmos.

What happens to $f$ at $x=1.5$?

The graph of $f'$ crosses the x-axis at $x=1.5$. It is negative before that point and positive after it, so $f$ is decreasing then increasing. That makes $x=1.5$ a relative minimum of $f$. This is just the First Derivative Test applied graphically.

Here are $f(x)=x^2-3x$ and $f'(x)=2x-3$ together so you can see the connection.

![Graph of f and f prime](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled_2.png-1704919445178-30848)

###### Graph of $f(x)=x^2-3x$ and its derivative $f'(x)=2x-3$. Image created with Desmos.

### Worked Example: Extrema and Inflection on One Graph

Returning to the graph of $g(x)$:

![Graph of g(x)](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled.png-1704919445168-25212)

###### Graph of function $g(x)$. Image courtesy of Zweig Media.

- At $x=-2$ and $x=2.8$, $g(x)$ has relative minima, so $g'(x)$ has x-intercepts there and changes from negative to positive.
- At $x=0.85$, $g(x)$ has a relative maximum, so $g'(x)$ has an x-intercept and changes from positive to negative.
- At $x=-0.5$, $g(x)$ has a point of inflection (concave up to concave down), so $g'(x)$ has a relative maximum and $g''(x)$ has an x-intercept changing from positive to negative.
- At $x=1.5$, $g(x)$ has a point of inflection (concave down to concave up), so $g'(x)$ has a relative minimum and $g''(x)$ has an x-intercept changing from negative to positive.

### Worked Example: Cubic Function

Look at the graph of $h(x)=x^3+2x^2$ and think about three things:

1. What happens to $h'(x)$ at $x=-1.3$ (red dotted line)?
2. What happens to $h'(x)$ at $x=-0.667$ (black dotted line)?
3. What happens to $h''(x)$ at $x=-0.667$?

![Graph of h(x)](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled_3.png-1704919445179-99711)

###### Graph of $h(x)=x^3+2x^2$. Image created with Desmos.

At $x=-1.3$, $h(x)$ has a relative maximum, so $h'(x)$ has an x-intercept there and changes from positive to negative.

At $x=-0.667$, $h(x)$ changes from concave down to concave up, so $h'(x)$ has a relative minimum and $h''(x)$ has an x-intercept changing from negative to positive.

Here is $h(x)$ in blue and $h'(x)$ in green so you can see those trends.

![Graph of h and h prime](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled_4.png-1704919445180-59297)

###### Graph of $h(x)=x^3+2x^2$ and $h'(x)=3x^2+4x$. Image created with Desmos.

And here is the same graph with $h''(x)$ added in purple.

![Graph of h, h prime, and h double prime](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled_5.png-1704919445180-81803)

###### Graph of $h(x)=x^3+2x^2$, $h'(x)=3x^2+4x$, and $h''(x)=6x+4$. Image created with Desmos.

### Practice Problems

**Question 1:**

The second derivative $f''$ of the differentiable function $f$ is graphed.

![Graph of f double prime](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled_6.png-1704919445181-28347)

###### Graph of the second derivative of $f$ with the point $(1, 6)$ labeled. Image created with Desmos.

Given that $f'(1)=0$, what can you tell about $f$ at $x=1$ based on the graph of $f''$?

**Question 2:**

The second derivative $f''$ of the differentiable function $f$ is graphed.

![Graph of f double prime](https://storage.googleapis.com/static.prod.fiveable.me/images/Untitled_7.png-1704919445182-25766)

###### Graph of the second derivative of $f$ with the point $(-1.8, 0)$ labeled. Image created with Desmos.

What can you tell about $f$ at $x=-2.4$ based on the graph of $f''$?

### Answers and Solutions

**Question 1:**

Answer: $f$ has a relative minimum at $x=1$.

[Solution](/ap-calc/unit-7/verifying-solutions-for-differential-equations/study-guide/s2nX7AhIBDxGIWwlL82x "fv-autolink"): The graph of $f''$ is positive at $x=1$, so $f$ is concave up there. Combined with $f'(1)=0$, the Second Derivative Test tells you $f$ has a relative minimum at $x=1$.

**Question 2:**

Answer: $f$ is concave down at $x=-2.4$.

Solution: The graph of $f''$ is negative at $x=-2.4$, so $f$ is concave down there.

## Common Misconceptions

- Treating features of the $f'$ graph as if they belong to $f$. If the graph of $f'$ is going up, that means $f$ is concave up, not that $f$ itself is increasing. Increasing $f$ comes from $f'$ being positive, not from $f'$ rising.
- Confusing an x-intercept of $f'$ with an [inflection point](/ap-calc/key-terms/inflection-point "fv-autolink"). An x-intercept of $f'$ where $f'$ changes sign is a relative extremum of $f$. Inflection points of $f$ line up with extrema of $f'$, where $f''$ changes sign.
- Assuming every zero of $f'$ is a maximum or minimum. If $f'$ touches the x-axis but does not change sign, $f$ has no extremum there.
- Assuming every zero of $f''$ is an inflection point. An inflection point requires $f''$ to actually change sign, not just equal zero.
- Forgetting that [critical points](/ap-calc/key-terms/critical-points "fv-autolink") can come from $f'$ being undefined, such as at a cusp or vertical tangent, not only from $f'=0$.
- Using vague words like "it" in justifications. Always name $f$, $f'$, or $f''$ so your reasoning is clear.

## Related AP Calculus Guides

- [Unit 5 Overview: Analytical Applications of Differentiation](/ap-calc/unit-5/review/study-guide/22AdFpcITnvM0bXKRl55)
- [5.1 Using the Mean Value Theorem](/ap-calc/unit-5/using-mean-value-theorem/study-guide/79sP2PXcyvRvBsjb3HRq)
- [5.2 Extreme Value Theorem, Global vs Local Extrema, and Critical Points](/ap-calc/unit-5/extreme-value-theorem-global-vs-local-extrema-critical-points/study-guide/xcQI1ZzNbmWJ5uRNiFCo)
- [5.3 Determining Intervals on Which a Function is Increasing or Decreasing](/ap-calc/unit-5/determining-intervals-on-which-function-is-increasing-or-decreasing/study-guide/Y2fgjyl7H1dKPI2YsB4Y)
- [5.4 Using the First Derivative Test to Determine Relative (Local) Extrema](/ap-calc/unit-5/using-first-derivative-test-to-determine-relative-local-extrema/study-guide/BjnQNCShz0uQiZGhSl2g)
- [5.11 Solving Optimization Problems](/ap-calc/unit-5/solving-optimization-problems/study-guide/u2Y3MpOG6kkTtbLH38S7)

## Vocabulary

- **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.

## FAQs

### How are f, f', and f'' connected in AP Calculus?

The first derivative f' tells you where f is increasing or decreasing. The second derivative f'' tells you where f is concave up or concave down. Features on one graph help you justify behavior on the others.

### How do you use f' to describe f?

When f' is positive, f is increasing. When f' is negative, f is decreasing. If f' changes from negative to positive, f has a relative minimum; if f' changes from positive to negative, f has a relative maximum.

### How do you use f'' to describe f?

When f'' is positive, f is concave up. When f'' is negative, f is concave down. A sign change in f'' can indicate a point of inflection on f.

### What does an extrema of f' mean for f?

A relative maximum or minimum of f' usually lines up with a point where the concavity of f changes. That means it can help identify an inflection point of f.

### What is the common mistake with graphs of f and f'?

The common mistake is treating the graph of f' as if it were the graph of f. If f' is increasing, that means f is concave up, not necessarily that f is increasing.

### How is AP Calculus 5.9 tested?

AP Calculus 5.9 is often tested with graph-based questions that ask you to justify increasing and decreasing intervals, extrema, concavity, or inflection points using f', f'', and sign changes.

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