Determining concavity — AP Calculus AB Study Guide

For: AP Calculus AB candidates sitting AP Calculus AB.

Covers: Formal definition of concave up and concave down, the second derivative rule for concavity, inflection point identification, concavity interpretation from first derivative graphs, and interval testing for concavity over a domain.

You should already know: How to compute first and second derivatives of differentiable functions; how to write and interpret open interval notation; how to test the sign of a function on an interval.

A note on the practice questions: All worked questions in the "Practice Questions" section below are original problems written by us in the AP Calculus AB style for educational use. They are not reproductions of past College Board / Cambridge / IB papers and may differ in wording, numerical values, or context. Use them to practise the technique; cross-check with official mark schemes for grading conventions.

1. What Is Determining concavity?

Concavity describes the direction of curvature of a differentiable function $f(x)$ over an interval, and it is a core high-frequency skill on the AP Calculus AB exam. Per the AP CED, Unit 5 (Analytical Applications of Differentiation) makes up 10–15% of total exam weight, and determining concavity appears in both multiple-choice (MCQ) and free-response (FRQ) sections. MCQ questions often ask to identify concavity from a graph of $f$ or $f^{\prime }$, while FRQ questions require justifying intervals of concavity or identifying inflection points as part of larger graph analysis or optimization problems.

Informally, concave up intervals are shaped like a cup ($\cup$) that "holds water", and concave down intervals are shaped like a cap ($\cap$) that "spills water". Formally, concavity is defined by the position of tangent lines relative to the function: a function is concave up on an interval if all tangent lines on the interval lie below the function, and concave down if all tangent lines lie above the function. Determining concavity is a prerequisite for identifying inflection points and using the second derivative test for local extrema, two other heavily tested Unit 5 skills.

2. The Second Derivative Rule for Concavity

Intuitively, concavity describes how the slope of the tangent line (the first derivative $f^{\prime }(x)$) changes as $x$ increases. If a function is concave up, its slope increases as $x$ increases: it starts negative, becomes less negative, then positive, growing steeper over the interval. If it is concave down, its slope decreases as $x$ increases. Because the second derivative $f^{\prime \prime }(x)$ measures the rate of change of $f^{\prime }(x)$, this gives us a direct, testable relationship between the sign of $f^{\prime \prime }$ and concavity:

Second Derivative Rule for Concavity: If $f$ is twice differentiable on an open interval $I$:

1. If $f^{\prime \prime }(x)>0$ for all $x\in I$, then $f$ is concave up on $I$.
2. If $f^{\prime \prime }(x)<0$ for all $x\in I$, then $f$ is concave down on $I$.

The standard process for finding intervals of concavity is: (1) compute $f^{\prime \prime }(x)$, (2) find all points where $f^{\prime \prime }(x)=0$ or $f^{\prime \prime }(x)$ is undefined (these split the domain of $f$ into open test intervals), (3) test the sign of $f^{\prime \prime }(x)$ in each interval, (4) assign concavity based on the sign of $f^{\prime \prime }$.

Worked Example

Find all intervals where $f(x)=x^3-6x^2+9x+2$ is concave up and concave down.

1. Compute the first derivative: $f^{\prime }(x)=3x^2-12x+9$.
2. Compute the second derivative: $f^{\prime \prime }(x)=6x-12=6(x-2)$.
3. Find candidate split points: $f^{\prime \prime }(x)$ is defined for all real $x$, and $f^{\prime \prime }(x)=0$ only at $x=2$. This splits the domain into two intervals: $(-\infty ,2)$ and $(2,\infty )$.
4. Test the sign of $f^{\prime \prime }$: For $x<2$, test $x=0$: $f^{\prime \prime }(0)=-12<0$, so $f$ is concave down on $(-\infty ,2)$. For $x>2$, test $x=3$: $f^{\prime \prime }(3)=6>0$, so $f$ is concave up on $(2,\infty )$.

Exam tip: On FRQ questions, you must explicitly reference the sign of the second derivative in your justification (e.g., "concave up on $(2,\infty )$ because $f^{\prime \prime }(x)>0$ for all $x$ in this interval") to earn full credit.

3. Inflection Point Identification

An inflection point is a point on the graph of $f$ where concavity changes from up to down or down to up. For an inflection point to exist at $x=c$, two non-negotiable conditions must be met: (1) $f(c)$ is defined (the point $(c,f(c))$ lies on the graph of $f$), and (2) the sign of $f^{\prime \prime }(x)$ (and thus concavity) changes across $x=c$.

A common student misconception is that inflection points only occur where $f^{\prime \prime }(c)=0$, and that all points with $f^{\prime \prime }(c)=0$ are inflection points. This is incorrect: inflection points can also occur where $f^{\prime \prime }(c)$ is undefined (as long as $f(c)$ exists and concavity changes), and $f^{\prime \prime }(c)=0$ does not guarantee a concavity change. For example, $f(x)=x^4$ has $f^{\prime \prime }(0)=0$, but $f^{\prime \prime }(x)$ is positive on both sides of $x=0$, so no inflection point exists there.

The process for finding inflection points is: after generating a list of candidate points (all $x=c$ where $f^{\prime \prime }(c)=0$ or $f^{\prime \prime }(c)$ undefined, and $f(c)$ is defined), test the sign of $f^{\prime \prime }$ on both sides of $c$ to confirm a sign change.

Worked Example

Find all inflection points of $f(x)=x^4-4x^3+6x^2$.

1. Compute the second derivative: $f^{\prime }(x)=4x^3-12x^2+12x$, so $f^{\prime \prime }(x)=12x^2-24x+12=12(x-1{)}^2$.
2. Generate candidate points: $f^{\prime \prime }(x)$ is defined for all real $x$, and $f^{\prime \prime }(x)=0$ only at $x=1$, so the only candidate is $x=1$.
3. Test for a concavity change: For $x<1$, test $x=0$: $f^{\prime \prime }(0)=12(1)=12>0$ (concave up). For $x>1$, test $x=2$: $f^{\prime \prime }(2)=12(1{)}^2=12>0$ (also concave up).
4. Conclusion: Concavity does not change at $x=1$, so there are no inflection points for this function.

Exam tip: AP exam questions almost always include a distractor that relies on the mistake of assuming all $f^{\prime \prime }(c)=0$ points are inflection points. Always confirm the sign change.

4. Determining Concavity from a Graph of $f^{\prime }(x)$

The AP exam frequently tests the skill of determining concavity when given only the graph of the first derivative $f^{\prime }(x)$, not an explicit formula for $f(x)$. To do this, we use the core relationship between $f^{\prime }$ and $f^{\prime \prime }$: $f^{\prime \prime }(x)$ is equal to the slope of the tangent line to the graph of $f^{\prime }(x)$ at $x$.

This gives us a simple rule: if $f^{\prime }(x)$ is increasing on an interval, its slope is positive, so $f^{\prime \prime }(x)>0$, so $f(x)$ is concave up. If $f^{\prime }(x)$ is decreasing on an interval, its slope is negative, so $f^{\prime \prime }(x)<0$, so $f(x)$ is concave down. Inflection points on $f(x)$ correspond exactly to local extrema (peaks or valleys) on the graph of $f^{\prime }(x)$, because that is where the slope of $f^{\prime }$ changes sign, meaning $f^{\prime \prime }$ changes sign.

Worked Example

The graph of $f^{\prime }(x)$, the first derivative of $f(x)$, is a parabola opening downward with vertex at $x=3$, crossing the x-axis at $x=1$ and $x=5$. What interval is $f(x)$ concave up on?

1. Recall: $f(x)$ is concave up when $f^{\prime }(x)$ is increasing, because the slope of $f^{\prime }$ equals $f^{\prime \prime }(x)$.
2. A downward-opening parabola increases to the left of its vertex and decreases to the right of its vertex. This is a property of all quadratic functions opening downward.
3. The vertex of this parabola is at $x=3$, so $f^{\prime }(x)$ is increasing on $(-\infty ,3)$ and decreasing on $(3,\infty )$.
4. Therefore, $f(x)$ is concave up on $(-\infty ,3)$.

Exam tip: Do not confuse the sign of $f^{\prime }(x)$ (which tells you if $f(x)$ is increasing or decreasing) with the slope of $f^{\prime }(x)$ (which tells you concavity of $f(x)$). Ignore the vertical position of $f^{\prime }$ when finding concavity.

5. Common Pitfalls (and how to avoid them)

• Wrong move: Stating $x=c$ is an inflection point only because $f^{\prime \prime }(c)=0$, without checking that concavity changes at $x=c$. Why: Students memorize that inflection points occur where $f^{\prime \prime }=0$, so they stop there and skip the required sign check. Correct move: After finding all candidate points, always test the sign of $f^{\prime \prime }$ on both sides of the candidate to confirm a sign change, and confirm $f$ is defined at $c$.
• Wrong move: Confusing the sign of $f^{\prime }(x)$ with the slope of $f^{\prime }(x)$ when finding concavity from an $f^{\prime }$ graph, e.g., saying $f$ is concave up when $f^{\prime }(x)>0$. Why: Students mix up the two uses of the first derivative, one for monotonicity of $f$ and one for concavity of $f$ via the slope of $f^{\prime }$. Correct move: When asked for concavity from an $f^{\prime }$ graph, only pay attention to whether $f^{\prime }$ is increasing or decreasing, not whether it is positive or negative.
• Wrong move: Using closed intervals (e.g., $(-\infty ,2]$) for intervals of concavity on FRQ answers. Why: Students assume endpoints should be included because $f$ is defined there, but concavity is defined for open intervals. Correct move: Always write intervals of concavity as open intervals on the AP exam, unless explicitly asked otherwise.
• Wrong move: Ignoring candidate inflection points where $f^{\prime \prime }(c)$ is undefined. Why: Students only look for where $f^{\prime \prime }(x)=0$ and forget that points where the second derivative does not exist can still be inflection points. Correct move: When finding candidates, always include all points in the domain of $f$ where $f^{\prime \prime }(x)=0$ or $f^{\prime \prime }(x)$ is undefined, then test each for a concavity change.
• Wrong move: Claiming a decreasing function cannot be concave up, e.g., saying $f(x)=e^{-x}$ is concave down because it is decreasing. Why: Students incorrectly associate "decreasing" with "concave down", confusing slope direction with curvature direction. Correct move: Always separate monotonicity (from $f^{\prime }$ sign) and concavity (from $f^{\prime \prime }$ sign): any combination of increasing/decreasing and concave up/down is possible.

6. Practice Questions (AP Calculus AB Style)

Question 1 (Multiple Choice)

The second derivative of a function $f$ is given by $f^{\prime \prime }(x)=(x+1)(x-3{)}^2$. For what values of $x$ does the graph of $f$ have an inflection point? A) $x=-1$ only B) $x=3$ only C) $x=-1$ and $x=3$ D) No inflection points

Worked Solution: First, identify candidate inflection points by setting $f^{\prime \prime }(x)=0$, which gives candidates at $x=-1$ and $x=3$. Next, test for a sign change across each candidate. For $x<-1$, $f^{\prime \prime }(x)$ is negative, and for $-1<x<3$, $f^{\prime \prime }(x)$ is positive, so concavity changes at $x=-1$, confirming it is an inflection point. For $3<x$, $f^{\prime \prime }(x)$ is still positive, so no concavity change at $x=3$. Only $x=-1$ is an inflection point. Correct answer: A.

Question 2 (Free Response)

Let $f(x)=2x^3-9x^2+12x+4$. (a) Find all intervals where $f(x)$ is concave up and concave down. Justify your answer. (b) Identify all $x$-coordinates of inflection points of $f(x)$. Justify your answer. (c) Given that $f^{\prime }(1)=0$ and $f^{\prime }(2)=0$, use concavity to classify $x=1$ and $x=2$ as local maxima or local minima.

Worked Solution: (a) Compute derivatives: $f^{\prime }(x)=6x^2-18x+12$, $f^{\prime \prime }(x)=12x-18=6(2x-3)$. Set $f^{\prime \prime }(x)=0$ to get $x=\frac{3}{2}$. Testing intervals: for $x<\frac{3}{2}$, $f^{\prime \prime }(x)<0$, so $f$ is concave down on $\left(-\infty ,\frac{3}{2}\right)$; for $x>\frac{3}{2}$, $f^{\prime \prime }(x)>0$, so $f$ is concave up on $\left(\frac{3}{2},\infty \right)$. Justification: concavity follows the sign of $f^{\prime \prime }(x)$. (b) The only candidate inflection point is $x=\frac{3}{2}$. $f^{\prime \prime }$ changes from negative to positive at this point, so concavity changes, so $x=\frac{3}{2}$ is the only $x$-coordinate of an inflection point. (c) By the second derivative test: $f^{\prime \prime }(1)=12(1)-18=-6<0$, so $f$ is concave down at $x=1$, meaning $x=1$ is a local maximum. $f^{\prime \prime }(2)=12(2)-18=6>0$, so $f$ is concave up at $x=2$, meaning $x=2$ is a local minimum.

Question 3 (Application / Real-World Style)

The monthly profit $P(t)$ (in thousands of dollars) for a small business $t$ months after launching a new product line is given by $P(t)=-0.1t^3+1.2t^2+3.5t$, for $0\le t\le 10$. Find the interval where the profit function is concave down, and interpret your result in context.

Worked Solution: First compute derivatives: $P^{\prime }(t)=-0.3t^2+2.4t+3.5$, $P^{\prime \prime }(t)=-0.6t+2.4$. Set $P^{\prime \prime }(t)=0$ to get $t=4$. Test the interval $4<t<10$: pick $t=5$, $P^{\prime \prime }(5)=-0.6(5)+2.4=-0.6<0$, so $P(t)$ is concave down on $(4,10)$. In context, this means that after 4 months, the rate of growth of profit is decreasing; even though profit may still increase, it increases more slowly each month than it did in the first four months.

7. Quick Reference Cheatsheet

8. What's Next

Now that you can determine concavity and identify inflection points, the next immediate step is applying this knowledge to the second derivative test for local extrema, which relies entirely on concavity to classify critical points. This topic is also the foundation for full curve sketching, where you combine information about monotonicity, concavity, and inflection points to draw accurate graphs of functions from derivative information. Across Unit 5, all analytical applications of differentiation build on concavity to connect derivative behavior to global function properties, which is tested heavily on the exam. Without mastering how to correctly determine concavity, you will not be able to earn full credit on graph analysis FRQ questions.

Second Derivative Test for Local Extrema Sketching Graphs of Functions Analyzing First Derivative Graphs