AP®MathsCollege BoardCalculus BCRevision NotesUnit 5: Analytical Applications of DifferentiationGraphs of Functions & Their DerivativesIncreasing & Decreasing Functions

Increasing & Decreasing Functions (College Board AP® Calculus BC): Revision Note

Jamie Wood

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on

College Board Calculus BCCollege Board Calculus AB

Increasing & decreasing functions

What are increasing and decreasing functions?

  • A non-constant function space f is increasing on the interval open square brackets a comma space b close square brackets if:

    • f open parentheses x subscript 1 close parentheses less or equal than f open parentheses x subscript 2 close parentheses whenever x subscript 1 less than x subscript 2

    • If f open parentheses x subscript 1 close parentheses less than f open parentheses x subscript 2 close parentheses whenever x subscript 1 less than x subscript 2, then the function is strictly-increasing

  • A non-constant function space f is decreasing on the interval open square brackets a comma space b close square brackets if:

    • f open parentheses x subscript 1 close parentheses greater or equal than f open parentheses x subscript 2 close parentheses whenever x subscript 1 less than x subscript 2

    • If f open parentheses x subscript 1 close parentheses greater than f open parentheses x subscript 2 close parentheses whenever x subscript 1 less than x subscript 2, then the function is strictly-decreasing

  • A constant function is neither increasing nor decreasing

How do I find where a function is increasing and decreasing using the first derivative?

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

    • If the rate of change is positive, the function is increasing

    • If the rate of change is negative, the function is decreasing

  • This means you can determine if a function is increasing or decreasing at a point using the first derivative

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

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

  • You can also find an interval where a function is increasing or decreasing

    • To find where the function is increasing,

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

    • To find where the function is decreasing,

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

What happens if the first derivative is zero?

  • If f apostrophe open parentheses a close parentheses equals 0 then there is a critical point at x equals a

    • The function could be increasing, decreasing, or neither at these points

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

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

      • Consider f open parentheses x close parentheses equals x cubed

      • f apostrophe open parentheses 0 close parentheses equals 0

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

      • Therefore, f is increasing at (0, 0)

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

      • Consider f open parentheses x close parentheses equals negative x cubed

      • f apostrophe open parentheses 0 close parentheses equals 0

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

      • Therefore, f is decreasing at (0, 0)

    • If the sign of the derivative changes on either side of the point, then f is neither increasing nor decreasing

      • Consider f open parentheses x close parentheses equals x squared

      • f apostrophe open parentheses 0 close parentheses equals 0

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

      • Therefore, f is neither increasing nor decreasing at (0, 0)

    • If the derivative is also zero on either side of the point, then f is constant on an interval

      • Therefore, f is neither increasing nor decreasing

Examiner Tips and Tricks

The definitions for where a function is increasing or decreasing include the endpoints, however the scoring guidelines for exam questions often allow the point to still be awarded if the endpoints are not included.

I.e. " f open parentheses x close parentheses is increasing for 1 less or equal than x less or equal than 5 " would receive the same points as " f open parentheses x close parentheses is increasing for 1 less than x less than 5 ".

This usually comes up as an MCQ so you do not need to worry about whether to use an open or closed interval.

How can I identify intervals where the function is increasing or decreasing by using the graph of the derivative?

  • Sketching a graph of both f open parentheses x close parentheses and f apostrophe open parentheses x close parentheses can help to identify where a function will be increasing or decreasing

    • On the graph of f open parentheses x close parentheses,

      • An upward slope from left to right is where the function is increasing

      • A downward slope from left to right is where the function is decreasing

    • On the graph of f apostrophe open parentheses x close parentheses,

      • The portion of the graph above the x-axis is where the function is increasing

      • The portion of the graph below the x-axis is where the function is decreasing

  • The diagram below shows a cubic and its derivative, a quadratic, plotted on the same graph

    • Between the critical points at a and b, the cubic is decreasing

    • Therefore, the graph of the derivative is below the x-axis between a and b

Graph showing a black curve y=f(x), a red dashed curve y=f'(x). Points a and b are marked on the x-axis where f(x) changes from increasing to decreasing and vice versa
Graph of a cubic, and its derivative; a quadratic.

Worked Example

Let space f be the function given by space f open parentheses x close parentheses equals 1 fourth x to the power of 4 plus 1 third x cubed minus 3 x squared plus 4. On which of the following intervals is the function space f decreasing?

(A) open parentheses negative infinity comma space minus 3 close parentheses and open parentheses 0 comma space 2 close parentheses

(B) left parenthesis negative 4.054 comma space minus 1.144 right parenthesis and left parenthesis 1.399 comma space 2.466 right parenthesis

(C) open parentheses negative 3 comma space 0 close parentheses and open parentheses 2 comma space infinity close parentheses

(D) left parenthesis negative 1.786 comma space 1.120 right parenthesis

Answer:

The function is decreasing where f apostrophe open parentheses x close parentheses less than 0

Find f apostrophe open parentheses x close parentheses

f apostrophe open parentheses x close parentheses equals x cubed plus x squared minus 6 x

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

table row cell x cubed plus x squared minus 6 x end cell less than 0 row cell x open parentheses x squared plus x minus 6 close parentheses end cell less than 0 row cell x open parentheses x plus 3 close parentheses open parentheses x minus 2 close parentheses end cell less than 0 end table

The easiest way to solve a cubic inequality is to graph it, you could use your calculator to help you

Graph of a cubic function crossing the x-axis at (-3, 0), (0, 0), and (2, 0), and the y-axis at (0, 0), with labeled coordinates.

Use the graph to identify where table row cell x open parentheses x plus 3 close parentheses open parentheses x minus 2 close parentheses end cell less than 0 end table (the parts underneath the x-axis)

x less than negative 3 and 0 less than x less than 2

So these are the regions where f apostrophe open parentheses x close parentheses less than 0, therefore these are the regions where the graph of f open parentheses x close parentheses is decreasing

The question asks for intervals, rather than values of x

Decreasing on the intervals open parentheses negative infinity comma space minus 3 close parentheses and open parentheses 0 comma space 2 close parentheses

The following options are incorrect:

  • (B) because these are the intervals where space f is negative

  • (C) because these are the intervals where space f is increasing

  • (D) because this is the interval where the derivative of space f is decreasing

Worked Example

Coordinate graph of y = f′, a smooth sinusoidal curve crossing the x-axis near x = 1, 3, 5 and 7, with peaks between 3–4 and 7–8 and troughs near 2 and 6.

The function space f is differentiable on the open interval open parentheses 0 comma space 8 close parentheses. The graph of space f apostrophe, the derivative of space f is shown in the figure above.

On what open intervals is space f increasing? Justify your answer.

Answer:

Check when space f apostrophe open parentheses x close parentheses greater than 0

space f apostrophe open parentheses x close parentheses greater than 0 when 0 less than x less than 1, 3 less than x less than 5 and 7 less than x less than 8

Therefore, space f is increasing on the intervals open parentheses 0 comma space 1 close parentheses, open parentheses 3 comma space 5 close parentheses and open parentheses 7 comma space 8 close parentheses because space f apostrophe open parentheses x close parentheses greater than 0 on these intervals

Examiner Tips and Tricks

For the worked example above, the function is not increasing at x equals 1 comma space 3 comma space 5 space or space 7 because the sign of space f apostrophe changes either side of these points.

You do not need to mention this in the answer because the question already asks for the open intervals.

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Author
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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.