AP®MathsCollege BoardCalculus ABRevision NotesUnit 5: Analytical Applications of DifferentiationGraphs of Functions & Their DerivativesCandidates Test for Global Extrema

Candidates Test for Global Extrema (College Board AP® Calculus AB)

Revision Note

Jamie Wood

Expertise

Maths

Candidates test

What is the candidates test for global extrema?

  • For a continuous function on a closed interval, the extreme value theorem guarantees

    • that there will be at least one global maximum and one global minimum

  • The global extrema (minimum or maximum) of a continuous function on a closed interval can only occur at:

    • Critical points

    • or at endpoints

  • This fact can be used to find global extrema using the following steps:

  • First check that f open parentheses x close parentheses is continuous on the interval open square brackets a comma space b close square brackets

  • Then find the critical points of f open parentheses x close parentheses on the interval open parentheses a comma b close parentheses

  • Find the value of f open parentheses x close parentheses at:

    • The critical points

    • The endpoints x equals a and x equals b

  • Out of these values,

    • the largest value of f open parentheses x close parentheses is the global maximum

    • the smallest value of f open parentheses x close parentheses is the global minimum

  • This process is known as the candidates test

    • The "candidates" for the global extrema are the critical points, and the endpoints

Exam Tip

  • Sketching a graph (or plotting on your calculator) is usually helpful

    • You may be able to spot immediately if an extremum will be at an endpoint or a critical point

  • A common mistake is to find critical points and assume they are global extrema

    • Even if the function is a quadratic, the critical point may not be the global extremum if the domain has been limited

Worked Example

Use the candidates test to find the coordinates of the global minimum and global maximum on the graph of the function f defined by:

f open parentheses x close parentheses equals negative x cubed plus 2 x squared plus 4 x minus 8 negative 3 less or equal than x less or equal than 3

Answer:

Check that f open parentheses x close parentheses is continuous on the interval

f open parentheses x close parentheses is continuous on [-3, 3] as it is a polynomial
(a cubic) defined for all values in the interval

Find the critical points of f open parentheses x close parentheses on the interval, where f to the power of apostrophe open parentheses x close parentheses equals 0

f to the power of apostrophe open parentheses x close parentheses equals negative 3 x squared plus 4 x plus 4

negative 3 x squared plus 4 x plus 4 equals 0

x equals 2 and x equals negative 2 over 3

Find the values of f open parentheses x close parentheses at the critical points

f open parentheses 2 close parentheses equals 0

f open parentheses negative 2 over 3 close parentheses equals negative 256 over 27 equals negative 9.481481...

Find the values of f open parentheses x close parentheses at the endpoints of the interval, in this case -3 and 3

f open parentheses negative 3 close parentheses equals 25

f open parentheses 3 close parentheses equals negative 5

Check the values of f open parentheses x close parentheses at the critical points and the endpoints to find the largest and smallest; these will be the global maximum and minimum respectively

The largest is 25, which occurs at an end point

The smallest is negative 256 over 27 which is -9.481481..., which occurs at a critical point

The global maximum is open parentheses negative 3 comma space 25 close parentheses

The global minimum is open parentheses negative 2 over 3 comma space minus 256 over 27 close parentheses

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    Jamie Wood

    Author: Jamie Wood

    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.