AP Syllabus focus:
‘All local extrema occur at critical points, but not every critical point is a local maximum or minimum, so each critical point must be checked carefully.’
Local extrema arise where a function’s derivative behaves in specific ways, and understanding how critical points relate to maxima and minima is essential for accurate function analysis.
Relating Critical Points to Local Extrema
This section explores how critical points indicate where a function’s graph may have a highest or lowest value relative to nearby points, and why careful verification is required. Critical points provide essential information about a function’s shape, behavior, and variation within an interval. The AP Calculus AB syllabus emphasizes that while all local extrema occur at critical points, not every critical point yields a local extremum. Therefore, developing a systematic approach for checking and interpreting these points is a central analytical skill.
Understanding Critical Points and Their Significance
A critical point is an -value in the domain of a function where the derivative is either zero or undefined. When equals zero, the tangent line is horizontal; when does not exist, the graph may exhibit a sharp turn or cusp. Each of these situations can indicate important structural changes in a graph.
Critical Point: A point in the domain of a function where or where does not exist.
A critical point represents a necessary condition for a local extremum, but it is not a sufficient condition. This means that a function cannot have a local maximum or minimum at any point that is not critical, yet some critical points do not correspond to peaks or valleys on the graph.
A polynomial curve displaying one local maximum and one local minimum, where the derivative is zero at each turning point. The labeled intervals show where the function rises or falls, illustrating how extrema relate to sign changes in slope. Source.
Between definition blocks and equation blocks, a normal explanatory sentence is required, ensuring clarity and conceptual flow for students.
Why Extremum Behavior Requires Verification
The derivative’s sign before and after a critical point determines whether the function is increasing or decreasing around that point. This sign information reveals whether the critical point represents a turning point or simply a change in slope behavior without forming a maximum or minimum. A local extremum arises only when the direction of change reverses.
A horizontal tangent alone does not imply a meaningful extremum. For example, a graph may flatten briefly but continue increasing afterward, meaning no local extremum occurs. Likewise, a derivative that does not exist may indicate a cusp with no maximum or minimum. Students must therefore avoid assuming an extremum solely from identifying a critical point.
Classifying Critical Points Conceptually
To determine whether a critical point corresponds to a local extremum, the behavior of the derivative must be analyzed. Key ideas guiding classification include:
A local maximum requires a change from increasing to decreasing.
A local minimum requires a change from decreasing to increasing.
No sign change in the derivative means no local extremum, even though the point remains critical.
Critical points that arise from an undefined derivative must still be checked using surrounding behavior.
This structure aligns with the overarching syllabus message: all local extrema occur at critical points, but not all critical points produce local extrema.
Using Derivative Information to Connect Critical Points and Extrema
The most direct way to evaluate critical points involves examining the sign of the first derivative. When changes sign around a critical point, the graph undergoes a directional shift that forms a local extremum. When the sign does not change, the graph continues its pattern of increasing or decreasing, leading to no extremum.
= First derivative describing the instantaneous rate of change
A sentence is required here between equation blocks to ensure the conceptual connection remains clear for students.
The first derivative reflects the underlying behavior of the function’s slope, allowing students to justify whether a maximum or minimum occurs at a critical point by referencing precise sign changes.
A cubic function with its first and second derivatives, showing where the derivative is zero and how the function’s slope and concavity behave. The marked stationary points correspond to potential extrema, while the second derivative curve adds context about concavity, which goes slightly beyond the scope of this subsubtopic but supports the analysis of turning points. Source.
Structural Features That Appear at Critical Points
Critical points often correspond to important visual features on a graph, such as peaks, valleys, or flat regions. However, some critical points describe subtler characteristics where the graph’s slope transitions gradually without yielding an extremum. Recognizing these nuances is essential, especially when connecting graphical and algebraic representations.
Bullet points help clarify common situations involving critical points:
Local Maxima
Occur at critical points.
Require to change from positive to negative.
Represent a high point relative to nearby values.
Local Minima
Occur at critical points.
Require to change from negative to positive.
Represent a low point relative to nearby values.
Neither Maximum Nor Minimum
Occurs when does not change sign.
Includes flat spots where the graph continues increasing or decreasing.
May involve undefined derivatives without turning behavior.
These layered distinctions help frame the essential idea: critical points are indicators, not guarantees. Students must analyze derivative behavior carefully to determine whether a critical point truly yields a local extremum.
FAQ
A point where the derivative is zero may simply indicate a momentary flattening of the graph rather than a turning point.
This occurs when the function continues increasing or continues decreasing on both sides of the point.
Such points often occur in functions with higher-order flattening, such as smooth plateaus or points of inflection with horizontal tangents.
A critical point must lie within the domain of the original function; otherwise, it cannot be considered a candidate for a local extremum.
For instance, if the derivative is undefined at a boundary point or where the function itself is not defined, the point cannot host a local maximum or minimum.
Ensuring domain membership prevents incorrect classification of behaviour at discontinuities or excluded values.
Yes. A local extremum may occur at a corner or cusp provided the function changes from increasing to decreasing or vice versa at that point.
Key details at such points include:
• The one-sided derivatives behave differently and do not form a smooth transition.
• The derivative fails to exist, but the function’s directional change still satisfies the requirement for an extremum.
The behaviour on intervals immediately adjacent to the critical point reveals whether the function rises or falls around that point.
A structured approach is:
• Determine the sign of the derivative on both sides.
• Compare the direction of change approaching and leaving the point.
• If the direction reverses, a local extremum exists; if not, it does not.
Yes. A function may have several critical points for reasons unrelated to turning behaviour.
Some critical points may correspond to:
• Flat points with no directional reversal
• Points where the derivative is undefined yet the function does not switch between increasing and decreasing
• Inflection points with horizontal tangents
Only those points where the derivative changes sign represent genuine local extrema.
Practice Questions
Question 1 (1–3 marks)
The derivative of a function f is given by the sign chart below:
• f'(x) > 0 for x < 1
• f'(x) = 0 at x = 1
• f'(x) < 0 for x > 1
(a) State whether f has a local maximum, a local minimum, or neither at x = 1.
(b) Give a brief reason for your answer based on the behaviour of f'(x).
Question 1
(a) 1 mark
• Local maximum at x = 1.
(b) 1–2 marks
• 1 mark for stating that the derivative changes from positive to negative.
• 1 mark for explaining that this corresponds to a change from increasing to decreasing, which indicates a local maximum.
Question 2 (4–6 marks)
A function g is differentiable on an open interval containing x = 3.
You are told the following information:
• g'(3) = 0
• g'(x) < 0 for x < 3
• g'(x) < 0 for x > 3
(a) Determine whether g has a local extremum at x = 3.
(b) Explain why the information about g'(x) implies your answer in part (a).
(c) Describe fully the behaviour of the function g on the interval and explain how this confirms your conclusion about whether x = 3 is an extremum.
Question 2
(a) 1 mark
• No local extremum at x = 3.
(b) 1–2 marks
• 1 mark for stating that g'(x) is negative on both sides of x = 3.
• 1 mark for recognising that there is no sign change in the derivative.
(c) 2–3 marks
• 1 mark for describing that g is decreasing for x < 3.
• 1 mark for stating g is also decreasing for x > 3.
• 1 mark for explaining that because there is no change from increasing to decreasing or from decreasing to increasing, x = 3 cannot be a local maximum or minimum.
