---
title: "AP Calculus 5.6: Determining Concavity"
description: "Review AP Calculus 5.6 determining concavity, including how f'', f' increasing or decreasing, sign charts, concave up and down intervals, and points of inflection work."
canonical: "https://fiveable.me/ap-calc/unit-5/determining-concavity/study-guide/ORBIficQDT458eUIhJ0V"
type: "study-guide"
subject: "AP Calculus AB/BC"
unit: "Unit 5 – Analytical Applications of Differentiation"
lastUpdated: "2026-06-11"
---

# AP Calculus 5.6: Determining Concavity

## Summary

Review AP Calculus 5.6 determining concavity, including how f'', f' increasing or decreasing, sign charts, concave up and down intervals, and points of inflection work.

## Guide

Concavity describes how a curve bends: [concave up](/ap-calc/key-terms/concave-up "fv-autolink") when it opens upward and [concave down](/ap-calc/key-terms/concave-down "fv-autolink") when it opens downward. You find concavity by checking the sign of the second derivative, where $f''(x)>0$ means concave up and $f''(x)<0$ means concave down. For AP Calculus, justify concavity with the sign of $f''$ or with whether $f'$ is increasing or decreasing.

## Why This Matters for the AP Calculus Exam

Concavity is one of the core ways AP Calculus asks you to analyze a [function](/ap-calc/unit-1/defining-continuity-at-point/study-guide/JbsR9iQfAzCznNOCG6JK "fv-autolink") using its derivatives. You will use the second derivative to describe how a graph bends, find [points of inflection](/ap-calc/key-terms/points-of-inflection "fv-autolink"), and justify those conclusions with clear reasoning. This shows up both in multiple-choice questions, where you read graphs and signs quickly, and in free-response questions, where you have to explain why a function is concave a certain way based on $f'$ or $f''$. Strong justification language here also sets you up for later topics like the second derivative test, sketching graphs, and connecting $f$, $f'$, and $f''$.

## Key Takeaways

- A graph is concave up on an interval when $f'$ is increasing, and concave down when $f'$ is decreasing.
- $f''(x)>0$ means concave up; $f''(x)<0$ means concave down.
- A possible point of inflection occurs where $f''(x)=0$ or where $f''$ is undefined.
- A point is an actual point of inflection only if $f''$ changes sign there, meaning the concavity really switches.
- Build a sign chart for $f''$ to find concavity over intervals, just like you do with $f'$ for increasing/decreasing.
- Always justify concavity by referring to $f''$ or to $f'$ increasing/decreasing, and name $f$, $f'$, and $f''$ specifically.

## Determining Concavity

A function is **concave up** when it opens upward and **concave down** when it opens downward. More precisely:

- If the slopes of the [tangent](/ap-calc/unit-2/derivatives-tangent-cotangent-secant-cosecant-functions/study-guide/wkBAhxiFZ1wV1M5mwLwt "fv-autolink") lines are increasing (that is, $f'$ is increasing), the function is concave up.
- If the slopes of the tangent lines are decreasing (that is, $f'$ is decreasing), the function is concave down.

### The Second Derivative

Since the second derivative is the derivative of the [first derivative](/ap-calc/key-terms/first-derivative "fv-autolink"), it tells you whether $f'$ is increasing or decreasing. That connects directly to concavity:

- A positive second derivative means $f'$ is increasing, so the function is concave up.
- A negative second derivative means $f'$ is decreasing, so the function is concave down.

The point where a function switches concavity is called an **[inflection point](/ap-calc/key-terms/inflection-point "fv-autolink")**. Because $f'$ changes from increasing to decreasing (or the reverse) there, $f''$ switches signs at that point. So if $f''(x)=0$ at a point, that point is a *possible* inflection point. You still have to confirm that the sign of $f''$ actually changes.

To sum up the key points:

1. A graph is concave up if $f'(x)$ is increasing, and concave down if $f'(x)$ is decreasing.
2. If $f''(x)>0$, the function is concave up. If $f''(x)<0$, the function is concave down.
3. A **point of inflection** is a point where $f$ changes concavity. If you only know $f''(x)=0$, you have a possible point of inflection, not a confirmed one.

### Concavity Walkthrough

Consider the function:

$$f(x)=x^3-3x^2+2x+1$$

a) Determine the intervals where $f(x)$ is concave up and concave down.

b) Find where $f(x)$ has points of inflection.

#### Walkthrough of Part A

To determine concavity, find the intervals where $f''(x)$ is positive (concave up) or negative (concave down).

Find the first and second derivatives using the [power rule](/ap-calc/unit-2/applying-power-rule/study-guide/GMr6EEbZezsP1DvqrpEk "fv-autolink").

$$f'(x)=3x^2-6x+2$$

$$f''(x)=6x-6$$

To understand the behavior of $f''(x)$, find a possible inflection point by setting $f''(x)=0$.

$$0=6x-6$$

$$6=6x$$

$$x=1$$

There is a possible point of inflection at $x=1$. This divides the [domain](/ap-calc/key-terms/domain "fv-autolink") into two intervals, $(-\infty,1)$ and $(1,\infty)$. Test a point in each interval to find the sign of $f''$.

Test $x=0$:

$$f''(0)=6(0)-6=-6$$

Since $f''(0)<0$, the interval to the left of $x=1$ is concave down.

Test $x=2$:

$$f''(2)=6(2)-6=6$$

Since $f''(2)>0$, the interval to the right of $x=1$ is concave up.

So $f(x)$ is concave down on $(-\infty,1)$ and concave up on $(1,\infty)$.

#### Walkthrough of Part B

Now check whether the possible point of inflection at $x=1$ is real. For that to be true, $f$ must change concavity at $x=1$.

Part A showed that it does, going from concave down to concave up. So $f(x)$ has a point of inflection at $x=1$, because $f$ changes concavity there and $f''(1)=0$.

## How to Use This on the AP Calculus Exam

### MCQ

For multiple-choice, you often read a graph of $f'$ or $f''$ and translate it into a statement about $f$. Remember that a graph of $f'$ that is increasing means $f$ is concave up, and a graph of $f''$ above the x-axis means $f$ is concave up. Watch for questions that hand you $f''$ and ask about concavity or inflection points directly, since you can answer those by checking signs.

### Free Response

For free-response, you usually have to find concavity and justify it. State the second derivative, identify where $f''=0$ or is undefined, build a sign chart, and report the intervals. Then justify clearly, for example: "$f$ is concave up on $1<x<\infty$ because $f''(x)>0$ there." When the question gives you a graph of $f'$, it is often easier to argue from $f'$ directly: "$f$ is concave up because $f'$ is increasing."

### Problem Solving

Use a clean sign chart for $f''$ the same way you do for $f'$. Find the candidates where $f''=0$ or is undefined, split the domain at those values, and test one point in each interval. An inflection point only counts if $f''$ changes sign, so always confirm the switch.

## Concavity Practice Problems

Question 1: Let $h(x)=5x^3$. What is the concavity of $h$ at $x=5$?

Question 2: Let $h(x)=3x^4+2x^3$. What is the concavity of $h$ at $x=-\frac{1}{3}$?

### Concavity Answers and Solutions

*Question 1:*

To evaluate concavity at a point, find the second derivative at that point.

$$h'(x)=15x^2$$

$$h''(x)=30x$$

$$h''(5)=150$$

Since $h''(5)=150$ is positive, $h$ is concave up at $x=5$.

*Question 2:*

To evaluate concavity at a point, find the second derivative at that point.

$$h'(x)=12x^3+6x^2$$

$$h''(x)=36x^2+12x$$

$$h''\left(-\frac{1}{3}\right)=36\cdot \left(-\frac{1}{3}\right)^2+12\cdot \left(-\frac{1}{3}\right)=0$$

Since $h''\left(-\frac{1}{3}\right)=0$, $x=-\frac{1}{3}$ might be a point of inflection. Check the second derivative on both sides to see if the sign changes. If it switches, it is a point of inflection.

| $x$ | $h''(x)$ | Concavity |
|---|---|---|
| -0.4 | 0.96 | Concave up |
| -0.3 | -0.36 | Concave down |

Since $f''$ switches sign, the concavity changes, so $x=-\frac{1}{3}$ is a point of inflection.

## Common Misconceptions

- Setting $f''(x)=0$ does not automatically give an inflection point. You must confirm that $f''$ actually changes sign there. A function can have $f''=0$ at a point and still keep the same concavity.
- Inflection points can also occur where $f''$ is undefined, not only where $f''=0$. Always include those candidates in your sign chart.
- Concave up does not mean increasing. A function can be decreasing while still concave up. Concavity is about how the [slope](/ap-calc/key-terms/slope "fv-autolink") changes, not whether the function is going up or down.
- Do not mix up the graph of $f'$ with the graph of $f$. If $f'$ is increasing, that tells you $f$ is concave up, not that $f$ is increasing.
- Be precise in your language. Say which function you mean by naming $f$, $f'$, or $f''$ instead of using "it," since a vague justification can lose meaning.

## 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

- **concave down**: A property of a function where the graph curves downward, occurring when the function's derivative is decreasing on an interval.
- **concave up**: A property of a function where the graph curves upward, occurring when the function's derivative is increasing on an interval.
- **points of inflection**: Points on the graph of a function where the concavity changes from concave up to concave down or vice versa.
- **second derivative**: The derivative of the first derivative, denoted f'', which describes the concavity of a function and indicates where it is concave up or concave down.

## FAQs

### How do you tell if a function is concave up or concave down?

Use the second derivative or the behavior of f prime. If f double prime is positive, f is concave up. If f double prime is negative, f is concave down.

### What does f prime increasing mean for concavity?

If f prime is increasing on an interval, then f is concave up on that interval. If f prime is decreasing, then f is concave down.

### How do you find intervals of concavity?

Find where f double prime equals zero or is undefined, split the domain at those values, and test the sign of f double prime on each interval.

### What is a point of inflection?

A point of inflection is a point where the graph of f changes concavity. A zero of f double prime is only a candidate unless the sign actually changes.

### What is the common mistake with concavity?

The common mistake is thinking concave up means increasing. Concavity describes how the slope changes, so a function can be decreasing and still concave up.

### How is AP Calculus 5.6 tested?

AP Calculus 5.6 is tested through second-derivative sign charts, graph interpretation, inflection point justification, and statements about f based on f prime or f double prime.

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