AP Syllabus focus:
‘A global (absolute) extremum is the highest or lowest function value on the entire interval, while a local (relative) extremum is only highest or lowest compared with nearby points.’
AP Calculus students must distinguish how functions achieve extreme values on intervals, connecting derivative behavior to both broad trends and localized behavior that occurs near specific points.
Understanding Extrema on an Interval
A function’s extreme values describe where it reaches particularly high or low outputs. These extremes are essential for analyzing graphs, interpreting change, and justifying conclusions about a function’s behavior on a given interval. The AP Calculus AB syllabus emphasizes the distinction between global (absolute) extrema and local (relative) extrema, and understanding this distinction allows students to describe behavior accurately and to apply derivative techniques appropriately.
Global (Absolute) Extrema
A global extremum is introduced as the single highest or lowest value that the function reaches on the entire interval under consideration. Because this extremum depends on the full interval, its identification requires considering all relevant points, including endpoints when the interval is closed.
Global (Absolute) Extremum: A function value that is the single highest or lowest on an entire interval.
Global extrema describe the overall behavior of a function and are critical when determining maximum or minimum quantities in applied contexts. They help characterize the broadest features of a function, reflecting where it attains its largest or smallest output values.
Local (Relative) Extrema
A local extremum focuses on a neighborhood rather than the whole interval. It identifies peak or valley behavior that holds only when compared to nearby points rather than the entire domain.
Local (Relative) Extremum: A function value that is higher or lower than all nearby values, but not necessarily extreme on the full interval.
Local extrema mark places where the function’s trend shifts direction. These points help highlight where the function transitions from increasing to decreasing or the reverse, revealing structural details of the graph.
Distinguishing Global and Local Extrema
Because the syllabus requires students to understand the difference clearly, it is important to break down how these extrema behave across the interval and how they relate to each other.
Scope of Comparison
Global extrema require comparing function values across the entire interval.
Local extrema require comparing function values only within a small surrounding region.
A global extremum may also be a local extremum, but a local extremum is not guaranteed to be global.
This diagram shows a function with several turning points and labels each as a local or global extremum. It illustrates that multiple local extrema can exist while only one point is the global maximum and one point is the global minimum. The image also implicitly uses the broader term “extreme values,” which is additional context beyond the syllabus requirement. Source.
This relationship depends on how the function behaves away from the local region.
Location on the Interval
A local extremum must occur at an interior point because it must be compared to values on both sides.
A global extremum can occur:
at an interior point
or at an endpoint, if the interval includes that endpoint
Endpoints can never be local extrema because neighborhoods cannot extend beyond the boundary of the interval, but they can still produce the absolute highest or lowest value for the entire interval.
Importance of Interval Type
Whether the interval is open, closed, or half-open influences the existence of extrema. Only points included in the interval can be considered. A function may fail to achieve a global extremum if the interval does not contain its boundary points. Local extrema, however, depend only on behavior within the interval and can occur even when global extrema do not.
Using Derivatives to Understand Extrema Behavior
Understanding how extrema form requires examining derivative behavior near critical points. While this subsubtopic does not cover the derivative tests themselves, it relies on interpreting extrema once such points are found.
How Local Extrema Relate to Critical Points
Critical points occur where the first derivative equals zero or does not exist, but these points are only candidates for local extrema. Once identified, they can be examined to determine if the function exhibits a peak or valley relative to nearby values. This characterization helps classify which interior points might be local extrema.
Global Extrema and Derivative Behavior
Global extrema may be found at:
critical points inside the interval
endpoints of the interval (if included)
Derivatives help identify critical points, but finding global extrema ultimately requires evaluating the function itself across all candidates. Because global extrema describe the entire interval’s behavior, derivative information alone is not sufficient without comparing function values directly.
Visual Interpretation
Graphical understanding is an essential skill emphasized in AP Calculus.
This graph displays several peaks and valleys, labeling some as local extrema and one as the global maximum. It visually reinforces the distinction between neighborhood-based extrema and extrema defined across the entire interval. The page also contains real-world graphs not used here, which include additional context outside the syllabus scope. Source.
When analyzing graphs:
Local extrema appear as peaks or valleys within the interior.
Global extrema are the tallest or lowest visible points on the entire portion of the graph that belongs to the interval.
A local extremum may look significant within a section of the graph but still fall short of the highest or lowest overall value.
Practical Importance of the Distinction
Distinguishing global from local extrema allows students to:
accurately describe a function’s structure
discuss behavior with proper mathematical precision
reason about applied problems where the absolute highest or lowest quantity is required
Bullet points help clarify common misunderstandings:
A point can be a local maximum but not the global maximum.
The global maximum might occur at an endpoint even if no local extremum exists there.
A function can have multiple local extrema but only one global extremum on a given interval.
This distinction reinforces how calculus links algebraic structure, graphical features, and contextual interpretation across an entire interval.
FAQ
A function defined on an open interval may have multiple local maxima and minima but fail to achieve a global extremum because the endpoints are not included, allowing the values of the function to approach but never reach its highest or lowest possible output.
This occurs frequently in curves that grow without bound or approach limiting values that are never attained within the interval.
A horizontal segment can contain infinitely many points where the derivative is zero, but:
• It may represent a local extremum only if the entire segment is higher or lower than neighbouring values.
• It may contain a global extremum if that constant value is the highest or lowest on the whole interval.
A horizontal segment does not automatically imply either type of extremum; comparison with nearby and global behaviour is essential.
Symmetry can help predict where extrema occur:
• Even symmetry (mirror symmetry about the y-axis) can cause extrema to appear symmetrically placed or at the axis itself.
• Odd symmetry constrains extrema so that if a local maximum exists at a point, a corresponding local minimum may exist at the symmetric point.
However, symmetry offers no guarantee of global behaviour unless the interval itself is symmetric.
A function can have more than one global maximum (or minimum) if it attains the same highest (or lowest) value at multiple x-values.
In such cases, each point where the function reaches that shared extreme value is classified as a global extremum.
Local extrema may still exist in addition to these, but the global classification is determined entirely by the equality of function values.
A discontinuity does not prevent a point from being a global extremum as long as the function is defined at that point and the value is the highest or lowest on the interval.
However, discontinuities can obstruct comparisons needed to determine local extrema because a neighbourhood might not exist on both sides of a point.
This makes continuity essential for local classification, but not strictly required for global classification.
Practice Questions
Question 1 (1–3 marks)
A function f is defined on the closed interval [−2, 4]. The function has a local maximum at x = 1 and a local minimum at x = 3. At x = −2, f(−2) = 7, and at x = 4, f(4) = −1.
State the x-value(s) at which the global maximum and global minimum of f occur on the interval [−2, 4], and justify your answer using the distinction between local and global extrema.
Mark scheme:
• 1 mark for correctly identifying that the global maximum occurs at x = −2 because f(−2) = 7 is greater than all other given values.
• 1 mark for correctly identifying that the global minimum occurs at x = 4 because f(4) = −1 is less than all other given values.
• 1 mark for explaining that local extrema describe behaviour only in a neighbourhood, whereas global extrema are determined by comparing function values across the entire interval, including endpoints.
Question 2 (4–6 marks)
A function g is continuous on the interval [0, 6]. It has a local maximum at x = 2 and a local minimum at x = 5. The function values at the endpoints are g(0) = 4 and g(6) = 3. Additional information shows that g(2) = 7 and g(5) = 1.
(a) Determine the global maximum and global minimum of g on [0, 6].
(b) Explain why neither the local maximum at x = 2 nor the local minimum at x = 5 is guaranteed to be a global extremum without evaluating the function at the interval endpoints.
(c) Briefly describe why endpoints cannot be local extrema.
Mark scheme:
• 1 mark for stating that the global maximum is g(2) = 7.
• 1 mark for stating that the global minimum is g(5) = 1.
• 1–2 marks for explaining that local extrema describe the highest or lowest values only compared with nearby points, so they must be compared with endpoint values to determine whether they are global.
• 1 mark for stating that endpoints cannot be local extrema because there is no neighbourhood on both sides of the point.
• 1 mark for a clear and coherent explanation demonstrating understanding of the distinction between global and local extrema.
