Critical Points

Critical numbers tell us where to investigate; they do not automatically prove that a maximum or minimum occurs.

Definition

A number $c$ in the domain of $f$ is a critical number when

or $f'(c)$ does not exist. The corresponding point $(c,f(c))$ is a critical point on the graph.

The input must belong to the original function's domain. A vertical asymptote is not a critical point because no point of the function exists there.

Why critical numbers matter

At a smooth local maximum or minimum inside an interval, the tangent is usually horizontal, so $f'(c)=0$. Corners, cusps, and vertical tangents can also create extrema while the derivative is undefined.

Critical numbers therefore form the candidate list for local and absolute extrema.

A systematic method

1. Determine the domain of $f$.
2. Compute $f'(x)$.
3. Solve $f'(x)=0$.
4. Find domain inputs where $f'$ is undefined.
5. Keep only values belonging to the domain of $f$.
6. Evaluate or classify the candidates as required.

Worked example

Further testing is needed to classify them.

Undefined derivative example

For $f(x)=|x|$, the derivative is $-1$ for $x<0$ and $1$ for $x>0$, but it does not exist at $x=0$. Since $0$ is in the function's domain, it is a critical number and corresponds to a local minimum.

This is why solving only $f'(x)=0$ can miss important points.

Domain exclusion example

For $f(x)=1/x$, the derivative $f'(x)=-1/x^2$ is undefined at $x=0$. However, $0$ is not in the domain of $f$, so it is not a critical number.

It is a domain boundary and vertical asymptote, which still matters for interval analysis but receives a different label.

Endpoints and absolute extrema

Endpoints of a closed interval are not usually called critical numbers under the interior definition, but they are candidates for absolute maximum or minimum.

The closed-interval method compares $f$ at all interior critical numbers and both endpoints. The largest output is the absolute maximum; the smallest is the absolute minimum.

Stationary points without extrema

A horizontal tangent does not guarantee a turn. For $f(x)=x^3$, $f'(0)=0$, but the function increases on both sides of $0$. The origin is a stationary inflection point, not a local maximum or minimum.

Use derivative signs or other tests for classification.

Critical values in context

In an optimization model, critical inputs may represent candidate dimensions, times, or production levels. Reject values outside the physical domain and compare endpoints when the feasible interval is closed.

Report both the optimizing input and resulting output with units.

Technology as a check

A graph can locate likely critical behaviour, but a shallow turn or narrow feature may be missed by the viewing window. Algebraic derivative analysis provides exact candidates.

Use technology to verify, not replace, domain reasoning.

Common mistakes

Calling every derivative zero a maximum or minimum. It is only a candidate.

Ignoring points where the derivative is undefined. Keep them if the function exists there.

Including an asymptote as a critical point. The input is not in the domain.

Forgetting interval endpoints in absolute-extrema problems. Compare them separately.

Reporting only input values when points are requested. Evaluate $f(c)$ too.

Quick self-check

• What is the original function's domain?
• Where is $f'=0$?
• Where is $f'$ undefined, and does $f$ exist there?
• Are interval endpoints also candidates for the task?
• Have candidates been classified rather than assumed?
• Are contextual restrictions and units included?

Related topics

• First Derivative Test
• Optimization
• Concavity
• Curve Sketching