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

Concavity of Functions (College Board AP® Calculus AB)

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Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

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 ifspace 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

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

    • Concave down ifspace 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

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

Diagram comparing concave up (left) and concave down (right) curves, highlighting tangent lines.

  • You can see from the diagram that;

    • At a local minimum a function is concave up

      • f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses is positive

    • At a local maximum a function is concave down

      • f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses is negative

  • A point where a graph changes in concavity,

    • from concave up to concave down or vice versa,

    • is called a point of inflection

  • Points of inflection will therefore always have a second derivative of zero

    • They can have any value for the first derivative

ib-aa-sl-5-2-5-point-of-inflection-diagram

  • Note that not every point with a second derivative of zero is a point of inflection

    • The concavity has to change as well

    • E.g. for f open parentheses x close parentheses equals x to the power of 4

      • f to the power of apostrophe apostrophe end exponent open parentheses 0 close parentheses equals 0

      • But x equals 0 is not a point of inflection

        • f to the power of apostrophe apostrophe end exponent open parentheses x close parentheses equals 12 x squared greater than 0 for all x not equal to 0

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 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 or equal 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 or equal than x less or equal than pi

Alternatively you may have been able to find this region by inspecting the graph of y equals sin space x (the original function)

Concave down on [0, π]

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    Author
    Reviewer
    Jamie Wood

    Author: Jamie Wood

    Expertise: Maths

    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

    Author: Dan Finlay

    Expertise: Maths 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.