What Is A Local Extremum? (Calculus Rules)

A local extremum is a nearby high or low point of a function: a local maximum is at least as high as nearby values, and a local minimum is at least as low. To find one, identify critical numbers, then test how the function changes around each point. A critical number alone does not prove an extremum.

Do you want to know where a quantity reaches a nearby high or low, without mistaking a flat point for a turning point? That is the central job of local extrema in calculus. They help describe how a function behaves near a particular input, whether you are studying a graph, a model, or an optimization problem.

I use a simple routine: define the domain, find candidate points, test them, and report both the input and its function value. Keeping those steps separate prevents a common error: assuming that every point where the derivative is zero must be a maximum or minimum.

Local Extrema: Definitions, Domain, and Candidate Points

A local extremum describes a function’s behavior near one input, not across its entire domain. A local maximum is at least as high as nearby function values; a local minimum is at least as low. The domain matters because only inputs where the function is defined can be considered.

What do local maximum and local minimum mean?

A function (f) has a local maximum at (x=c) if (f(c)\ge f(x)) for all domain values (x) sufficiently close to (c). It has a local minimum if (f(c)\le f(x)) for those nearby values. These definitions compare nearby outputs, not every output.

The words “sufficiently close” mean that we examine a small neighborhood around (c). We do not need to choose a specific distance. The key is that the comparison holds for nearby inputs in the function’s domain.

Some textbooks use “relative maximum” and “relative minimum” to mean the same thing as local maximum and local minimum. A strict local maximum or minimum uses a stronger comparison: nearby values must be strictly lower or strictly higher, except at (c) itself.

Local versus absolute extrema

A local extremum is highest or lowest only in a nearby region. An absolute, or global, extremum is highest or lowest across the entire domain or interval being studied. A point can be a local maximum without being the function’s absolute maximum.

For example, a graph may have several peaks. Each peak can be a local maximum, but only the tallest peak is the absolute maximum, if one exists. When a question asks for local extrema, do not compare the entire graph unless the question also asks for absolute extrema.

What counts as a critical number?

A critical number is an input (c) where (f(c)) exists and either (f'(c)=0) or (f'(c)) does not exist. Critical numbers are candidates for local extrema, not proof of them. Check the domain first, because a point outside the domain cannot be a critical number of the function.

A derivative can be zero at a flat point that is not a peak or valley. It can also fail to exist at a corner where a local extremum does occur. That is why a complete search includes both kinds of candidates.

For instance, (f(x)=|x|) has a local minimum at (x=0). The function is defined there, but its derivative does not exist at the sharp corner. Looking only for solutions to (f'(x)=0) would miss this minimum.

Derivative Tests and Calculus Requirements

Derivative tests help classify critical numbers by describing what the function does nearby. The first-derivative test tracks whether the function rises or falls across a candidate. The second-derivative test uses concavity when its conditions apply. Neither test removes the need to check the domain.

Use Fermat’s theorem to find candidates

Fermat’s theorem says that if a function has a local extremum at an interior point (c), and is differentiable there, then (f'(c)=0). This gives a useful way to find candidates. It does not say that every point with a zero derivative is an extremum.

The assumptions matter. The point must be inside the domain, the function must have a local extremum there, and the function must be differentiable there. A local extremum at a corner can have an undefined derivative, as with (|x|) at zero.

A practical candidate search is:

• Find the function’s domain.
• Differentiate the function.
• Solve (f'(x)=0) for points in the domain.
• Add domain points where (f'(x)) does not exist.
• Test each candidate’s behavior.

Apply the first-derivative test

The first-derivative test checks the sign of (f’) just to the left and right of a critical number. A positive derivative means the function is increasing; a negative derivative means it is decreasing. A change from positive to negative signals a local maximum.

If the sign does not change, the function keeps moving in the same direction through the candidate, so that point is not a local extremum. The test can also classify a point where the derivative is undefined, as long as the function itself is defined and the nearby signs can be checked.

Use the second-derivative test when it applies

The second-derivative test classifies a critical point where (f'(c)=0). If (f”(c)>0), the graph is concave up there and the point is a local minimum. If (f”(c)<0), it is concave down and the point is a local maximum.

If (f”(c)=0), the test gives no classification. Do not call the point an extremum based on this result. Use the first-derivative test or another valid method instead.

Check endpoints separately

An endpoint of a closed interval is not an interior point, so Fermat’s theorem does not apply there in the usual way. If a problem asks for absolute extrema on a closed interval, evaluate the function at the endpoints and at every critical number inside the interval.

Some courses count an endpoint as a local extremum using comparisons within the domain; others reserve “local extremum” for interior points. Follow the definition used in your course. For an absolute-extrema question, the endpoint check is required regardless.

Worked Analysis, Common Errors, and Study Practice

A worked analysis turns the rules into a repeatable process: find candidate inputs, classify them, and calculate the corresponding outputs. The examples below show a smooth turning point, a flat point that is not an extremum, and a corner minimum. Each illustrates why testing matters.

Example 1: Find and classify extrema of (f(x)=x^3-3x)

Start with the domain: all real numbers. Differentiate:

[ f'(x)=3x^2-3=3(x-1)(x+1). ]

The critical numbers are (x=-1) and (x=1). Check the derivative’s sign:

• For (x<-1), (f'(x)>0), so the function increases.
• For (-1<x<1), (f'(x)<0), so the function decreases.
• For (x>1), (f'(x)>0), so the function increases.

The sign changes from positive to negative at (-1), giving a local maximum. Since (f(-1)=2), the point is ((-1,2)). At (1), the sign changes from negative to positive, giving a local minimum. Since (f(1)=-2), the point is ((1,-2)).

Example 2: A zero derivative that is not an extremum

Consider (g(x)=x^3). Its derivative is (g'(x)=3x^2), which is zero at (x=0). But (g'(x)) is positive on both sides of zero, so the function increases through the point. Therefore, (x=0) is a critical number, but not a local maximum or minimum.

The second derivative is (g”(x)=6x), so (g”(0)=0). That result is inconclusive, as the second-derivative test requires. The first-derivative test correctly shows that there is no extremum.

Example 3: A minimum where the derivative is undefined

For (h(x)=|x|), the function is defined at zero, but the derivative is not. To the left of zero the function decreases; to the right it increases. That change from negative to positive means (x=0) is a local minimum, with value (h(0)=0).

This case shows why the candidate list must include points where the derivative does not exist, provided the function itself exists there.

Common errors to avoid

• Calling every critical number an extremum: A critical number is only a candidate. Check derivative signs or use a conclusive second-derivative test.
• Ignoring the domain: A point where the function is undefined cannot be a local extremum of that function.
• Using the second-derivative test when (f”(c)=0): That result is inconclusive, not evidence for either classification.
• Reporting only the input: State the classification and the function value, such as “local maximum at ((-1,2)).”
• Forgetting interval endpoints: Include endpoints when finding absolute extrema on a closed interval.

Applications of Extrema in Optimization and Related Careers

Optimization uses maxima and minima to answer questions such as which allowed input produces the greatest or least modeled output. Calculus can locate candidates, but the function, domain, and constraints must accurately represent the problem. The same reasoning appears in many quantitative fields.

A model might describe cost, distance, output, or another measurable quantity as a function of an input. A local extremum identifies a nearby best or worst value in that model. It does not, by itself, prove that the result is best across every possible input.

For a real problem, I separate the math into three checks:

• Meaning: What does the input represent, and what does the function value measure?
• Allowed range: Which inputs are valid under the problem’s constraints?
• Conclusion: Is the question asking for a local or absolute result?

For example, if a model is only valid for (0\le x\le 10), a critical number outside that interval is irrelevant. If the task asks for the greatest value on that closed interval, compare function values at all interior critical numbers and both endpoints.

A reliable reporting format

A clear solution lists each candidate, the test used, and the result. For example: “At (x=1), (f’) changes from negative to positive, so (f) has a local minimum. Since (f(1)=-2), the minimum point is ((1,-2)).”

This format makes your reasoning easy to check. It also keeps the input (x), the output (f(x)), and the classification distinct.

FAQ: Local Extrema Rules

These short answers review the definitions and tests most often used to identify local maxima and minima. The key distinction is between finding a candidate and proving its classification. For any problem, start with the domain and follow the definition and conventions used in your course.

What is a local extremum?
A local extremum is a local maximum or local minimum: a point whose function value is at least as high or low as nearby values.

How do you find local extrema using derivatives?
Find the domain, solve (f'(x)=0), and include points where (f’) is undefined but (f) exists. Then classify each candidate with a derivative test.

Does (f'(c)=0) mean there is a local extremum?
No. A zero derivative makes (c) a candidate, but the function may keep increasing or decreasing through it.

What does a positive-to-negative derivative change mean?
It means the function changes from increasing to decreasing, so the point is a local maximum.

What does a negative-to-positive derivative change mean?
It means the function changes from decreasing to increasing, so the point is a local minimum.

Can a local extremum occur where the derivative does not exist?
Yes. If the function is defined there, a corner or cusp may be a local extremum. The function (|x|) has a minimum at zero.

What if the second derivative is zero?
The second-derivative test is inconclusive. Use the first-derivative test or another method to classify the point.

What is the difference between a local and absolute extremum?
A local extremum is highest or lowest nearby. An absolute extremum is highest or lowest across the full stated domain or interval.

Do endpoints count as local extrema?
Course definitions vary. Check whether your course allows a one-sided neighborhood within the domain. For absolute extrema on a closed interval, evaluate both endpoints.

What should I report for a local extremum?
Give the input, the classification, and the function value. For example, report a local minimum at ((1,-2)), not just (x=1).

(This article was written by one of our staff writers, Nathan Sterling. Visit our Meet the Team page.)