Critical points are essential in AP Calculus AB because they signal where a function’s behavior may change, allowing us to investigate possible maxima, minima, or flat regions in its graph.

Understanding the Meaning of Critical Points

A critical point is a location on the graph of a differentiable or piecewise-defined function where meaningful changes in behavior may occur. In calculus, these points help determine where a function may have local extrema, places where the function reaches a relative highest or lowest value. The AP specification emphasizes that critical points rely solely on information from the first derivative, because the derivative captures instantaneous rates of change along the graph.

A function must be defined at the point in question, meaning the $x$-value must lie within the domain of the original function. If the function does not exist at an $x$-value, that point cannot be considered a critical point even if the derivative has unusual behavior there.

Why Derivatives Identify Critical Points

The derivative $f'(x)$ measures the instantaneous rate of change of a function. When the derivative equals zero, the tangent line is horizontal. This situation is frequently—though not always—associated with local extrema. Conversely, when the derivative does not exist but the function remains defined, the graph often exhibits sharp corners, cusps, or vertical tangents. Each of these features marks a point where the behavior of the function could shift significantly.

Between intervals where the derivative is positive or negative, changes in sign must occur at critical points. Therefore, identifying such points is the foundation for analyzing increasing and decreasing behavior, curvature, and extremal values.

Formal Conditions for Identifying Critical Points

To determine whether an $x$-value qualifies as a critical point, two essential conditions guide the process:

• The function must be defined at the point.

• The first derivative must either

  • equal zero, or

  • fail to exist at that point.

These requirements ensure that critical points are tied to actual, meaningful features of the graph.

A common misconception is that any point where the derivative does not exist automatically becomes a critical point. However, if the function itself is not defined there—such as at a discontinuity or a vertical asymptote—the point cannot be critical because it fails the domain requirement.

Types of Situations That Produce Critical Points

Critical points arise from several structural characteristics of a function. Understanding these situations helps students recognize critical points more efficiently.

1. Horizontal Tangents

Horizontal tangents occur when $f'(x)=0$.
Key features include:

• Flat regions on the graph

• Turning points where direction may shift

• Plateaus where the slope temporarily becomes zero

Horizontal tangents are common sources of local maxima or minima, though not guaranteed indicators.

This graph shows a function with several local and global extrema, each occurring at a point where the tangent line is horizontal. It illustrates how a zero derivative may correspond to a maximum or minimum. The specific function $\cos(3\pi x)/x$ includes more complexity than required in AP Calculus AB, serving only as a concrete visual example. Source.

2. Nondifferentiable Points

A derivative may fail to exist even when the function is defined. These points often correspond to distinct geometric features.
Common nondifferentiable cases include:

• Cusps, where the graph sharply changes direction

• Corners, where one‐sided derivatives do not match

• Vertical tangents, where the slope grows without bound

In all these situations, the derivative is undefined, yet the function’s defined nature at the point qualifies the location as a critical point.

This diagram illustrates a cusp where the function is continuous and defined, yet the derivative fails to exist because the slopes from each side differ infinitely. Such points are nondifferentiable but still qualify as critical points. The image includes more detail than required, such as extended graph arrows, but the cusp feature is the key concept relevant to the syllabus. Source.

3. Points Involving Piecewise Functions

Piecewise functions require special attention. Even if each piece is differentiable on its own interval, the junctions may cause nondifferentiability. When evaluating such points, ensure:

• The pieces meet at the domain boundary

• The function is defined at the junction

• The derivative from each side either disagrees or one side fails to exist

If these conditions hold, the point becomes a critical point, making piecewise-defined functions a rich source of nondifferentiable behavior.

How Critical Points Fit into Broader Function Analysis

Identifying critical points is the first analytical step that precedes later techniques in calculus. These points serve as anchors for deeper investigations because they often signal meaningful transitions in a function’s graph.

After locating critical points, students use them to:

• Construct sign charts for the first derivative

• Determine intervals of increase or decrease

• Apply the First Derivative Test for classifying local extrema

• Connect behavior in $f$, $f'$, and $f''$

• Implement optimization methods in applied contexts

Critical points do not guarantee extremal values, but no local maxima or minima can occur without one. Thus, finding critical points isolates the limited set of candidates where more sophisticated tests can be applied.

Clarifying What Critical Points Are Not

To avoid common errors, it is important to distinguish true critical points from other noteworthy features.

• Discontinuities are never critical points because the function is not defined there.

• Endpoints of a closed interval are not considered critical points, even though they may produce extrema; they simply lie outside the derivative-based definition.

• Zeros of the function (where $f(x)=0$) are irrelevant unless accompanied by derivative conditions; they do not inherently indicate slope behavior.

Understanding these distinctions ensures that students use precise mathematical reasoning and remain aligned with the AP standard that emphasizes domain-based evaluation when determining critical points.