AP®MathsCollege BoardCalculus BCStudy GuidesUnit 5: Analytical Applications of DifferentiationGraphs of Functions & Their DerivativesConcavity of Functions

Concavity of Functions (College Board AP® Calculus BC): Study Guide

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

College Board Calculus BCCollege Board Calculus AB

Concavity of functions

What is concavity?

  • Concavity is the way in which a curve bends and is related to the second derivative of a function

  • A curve is:

    • Concave up if f to the power of apostrophe open parentheses x close parentheses is increasing in this interval

      • space f to the power of apostrophe apostrophe end exponent left parenthesis x right parenthesis greater or equal than 0 for all values ofspace x in an interval

    • Concave down if f to the power of apostrophe open parentheses x close parentheses is decreasing in this interval

      • space f to the power of apostrophe apostrophe end exponent left parenthesis x right parenthesis less or equal than 0 for all values ofspace x in an interval

Diagram comparing concave up (left) and concave down (right) curves, highlighting tangent lines.
Example of how tangents change based on the concavity of the curve

Examiner Tips and Tricks

Zero is included in both the inequalities above because the function could be concave up or concave down when space f apostrophe apostrophe open parentheses x close parentheses equals 0. However, in your exam, you will only need to use space f apostrophe apostrophe open parentheses x close parentheses greater than 0 or space f apostrophe apostrophe open parentheses x close parentheses less than 0 to find the concavity of functions. The functions will be chosen to make this work.

How do I find where a function is concave up or concave down using the second derivative?

  • The second derivative of a function, f apostrophe apostrophe open parentheses x close parentheses, describes the rate of change of the first derivative, f apostrophe open parentheses x close parentheses

    • If the rate of change is positive, the first derivative is increasing

    • If the rate of change is negative, the first derivative is decreasing

  • This means you can determine if a function is concave up or down at a point using the second derivative

    • If f apostrophe apostrophe open parentheses a close parentheses greater than 0 then f is concave up at x equals a

    • If f apostrophe apostrophe open parentheses a close parentheses less than 0 then f is concave down at x equals a

  • You can also find an interval where a function is concave up or down

    • To find where the function is concave up,

      • Solve the inequality f apostrophe apostrophe open parentheses x close parentheses greater than 0

    • To find where the function is concave down,

      • Solve the inequality f apostrophe apostrophe open parentheses x close parentheses less than 0

What happens if the second derivative is zero?

  • If f apostrophe apostrophe open parentheses a close parentheses equals 0 then the function could be concave up, concave down, or neither at these points

  • You determine what is happening at this point by looking at the sign of the second derivative on either side of this point

    • If the second derivative is positive on either side of the point, then f is concave up at x equals a

      • Consider f open parentheses x close parentheses equals x to the power of 4

      • f apostrophe apostrophe open parentheses 0 close parentheses equals 0

      • f apostrophe apostrophe open parentheses negative 0.01 close parentheses greater than 0 and f apostrophe apostrophe open parentheses 0.01 close parentheses greater than 0

      • Therefore, f is concave up at (0, 0)

    • If the second derivative is negative on either side of the point, then f is concave down at x equals a

      • Consider f open parentheses x close parentheses equals negative x to the power of 4

      • f apostrophe apostrophe open parentheses 0 close parentheses equals 0

      • f apostrophe apostrophe open parentheses negative 0.01 close parentheses less than 0 and f apostrophe apostrophe open parentheses 0.01 close parentheses less than 0

      • Therefore, f is concave down at (0, 0)

    • If the sign of the second derivative changes on either side of the point, then f is neither concave up nor concave down

      • Consider f open parentheses x close parentheses equals x cubed

      • f apostrophe apostrophe open parentheses 0 close parentheses equals 0

      • f apostrophe apostrophe open parentheses negative 0.01 close parentheses less than 0 and f apostrophe apostrophe open parentheses 0.01 close parentheses greater than 0

      • Therefore, f is neither concave up nor concave down at (0, 0)

      • It is called a point of inflection

How do I find points of inflection?

  • Find the points where the second derivative is zero

    • The value of the first derivative could be any value

  • Find the sign of the second derivative around the points to find whether the concavity changes

    • Check whether it goes from concave up to concave down or vice versa

    • If it does, then the point is a point of inflection

Graph of y = f(x) with concave down red sections, concave up green sections, and two blue points of inflection where fʺ(x) changes sign and equals zero
Examples of points of inflection

Examiner Tips and Tricks

In an exam, an easy way to remember the difference is:

  • Concave down is the shape of (the mouth of) a sad smiley ☹️

    • They are feeling negative!

  • Concave up is the shape of (the mouth of) a happy smiley 🙂

    • They are feeling positive!

Worked Example

The function f is defined by

f open parentheses x close parentheses equals sin x comma space space space 0 less or equal than x less or equal than 2 pi

State the open interval for which f is concave down.

Answer:

A function is concave down when f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses is negative

table row cell f to the power of apostrophe open parentheses x close parentheses end cell equals cell cos x end cell row cell f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses end cell equals cell negative sin x end cell end table

The following must then be solved in the given domain

negative sin x space less than space 0 comma space space space 0 less or equal than x less or equal than 2 pi

The easiest way to solve this is with a graph of y equals negative sin x

Sketch the graph of y equals negative sin x for 0 less or equal than x less or equal than 2 pi and highlight where the graph is less than zero

Graph of -sinx with 0 to pi highlighted (where it is below the x-axis)

The function is concave down on the interval where the second derivative is less than or equal to zero,

Concave down when 0 less than x less than pi

Concave down on open parentheses 0 comma space pi close parentheses because space f apostrophe apostrophe open parentheses x close parentheses less than 0

Worked Example

Let the function f be defined by f open parentheses x close parentheses equals x cubed minus 3 x squared plus 3.

Find the coordinates of the point of inflection on the graph of y equals f open parentheses x close parentheses. Justify how you know it is a point of inflection.

Answer:

Points of inflection have a second derivative of zero

Potential points of inflection can therefore be found by setting f apostrophe apostrophe open parentheses x close parentheses equals 0

f apostrophe apostrophe open parentheses x close parentheses equals 6 x minus 6

table row cell 6 x minus 6 end cell equals 0 row x equals 1 end table

Find the y-coordinate

f open parentheses 1 close parentheses equals 1

(1, 1)

So the point at (1, 1) could be a point of inflection

Check the sign of the second derivative at either side of x equals 1 to find the concavity of the curve

f apostrophe apostrophe open parentheses 0.9 close parentheses equals negative 0.6

f apostrophe apostrophe open parentheses 1.1 close parentheses equals 0.6

Just before x equals 1, space f is concave down because space f apostrophe apostrophe open parentheses x close parentheses less than 0

Just after x equals 1, space f is concave up because space f apostrophe apostrophe open parentheses x close parentheses greater than 0

Therefore, space f has a point of inflection at open parentheses 1 comma space 1 close parentheses because the concavity changes

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Jamie Wood

Author: Jamie Wood

Expertise: Maths Content Creator

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.

Dan Finlay

Reviewer: Dan Finlay

Expertise: Maths Subject Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.