The First Derivative Test for Local Extrema

What the First Derivative Test Is For

The first derivative tells you whether a function is increasing or decreasing. That makes it possible to determine what happens at a critical point by looking at how the sign of the derivative changes on either side.

If a function changes from increasing to decreasing, it has reached a local maximum. If it changes from decreasing to increasing, it has reached a local minimum.

The First Derivative Test turns that idea into a systematic procedure. Instead of trying to recognize the shape of the graph, you can calculate the sign of on each side of a critical point and let the sign change tell you the answer.

The Key Idea

Suppose is a critical number of . The important question is not simply whether . What matters is what the derivative does before and after the critical number.

There are three important possibilities:

Positive to negative: the function changes from increasing to decreasing, so there is a local maximum.
Negative to positive: the function changes from decreasing to increasing, so there is a local minimum.
No sign change: the function does not switch from increasing to decreasing or vice versa, so there is no local extremum at that critical number.

The First Derivative Test

For a critical number , use the following procedure:

Find the critical numbers. Solve and also check for points where does not exist, provided the function itself is defined there.
Place the critical numbers on a number line. These points divide the domain into intervals.
Determine the sign of on each interval. Choose one test point from each interval and substitute it into the derivative.
Look for a sign change. Positive-to-negative means local maximum; negative-to-positive means local minimum.
Find the corresponding function value if the problem asks for the local maximum or minimum point.

Worked Examples

Example 1: Local Maximum and Local Minimum

Use the First Derivative Test to find the local extrema of .

Step 1 — Find the derivative:

Step 2 — Find the critical numbers: Set the derivative equal to zero: . Therefore, the critical numbers are .

Step 3 — Test the intervals: The critical numbers divide the number line into .

Step 4 — Determine the signs:

Step 5 — Interpret the sign changes: At , the derivative changes from positive to negative, so the function changes from increasing to decreasing. Therefore, .

At the second critical number: At , the derivative changes from negative to positive, so .

Step 6 — Find the function values:

Result:

Example 2: A Critical Point That Is Not a Local Extremum

Use the First Derivative Test to determine whether has a local extremum at .

Step 1 — Find the derivative:

Step 2 — Find the critical number: , so .

Step 3 — Test the intervals: Choose one number less than 0 and one number greater than 0. For example, and .

Step 4 — Check the derivative signs:

Step 5 — Interpret: The derivative is negative on both sides of . There is no sign change, so the function is decreasing through the critical point.

Result:

Example 3: Factoring the Derivative

Use the First Derivative Test to find the local extrema of .

Step 1 — Differentiate:

Step 2 — Find critical numbers: , so the critical numbers are .

Step 3 — Determine signs: Because the derivative is already factored, evaluate the factors on each interval. The resulting sign pattern is .

Step 4 — Interpret: At , the derivative changes from positive to negative, so there is a local maximum. At , it changes from negative to positive, so there is a local minimum.

Step 5 — Find the function values:

Result:

Example 4: A Critical Number Where the Derivative Does Not Exist

Use the First Derivative Test to analyze .

Step 1 — Differentiate:

Step 2 — Find the critical number: The derivative is undefined when . The original function is defined there, so is a critical number.

Step 3 — Check the derivative signs: For , the derivative is negative. For , the derivative is positive.

Step 4 — Interpret: The derivative changes from negative to positive, so the function changes from decreasing to increasing.

Result:

How to Read a Sign Chart

A sign chart is often the fastest way to organize a First Derivative Test problem. Suppose the critical numbers are . You could summarize the derivative information like this:

Reading from left to right:

means the function is increasing.
means the function is decreasing.
The change from to identifies a local maximum.
The change from to identifies a local minimum.

First Derivative Test vs. Second Derivative Test

The First Derivative Test and Second Derivative Test are two different ways to classify critical points.

The First Derivative Test looks at whether changes sign.
The Second Derivative Test looks at the value of at a critical point.

The First Derivative Test is especially useful when the derivative factors nicely or when the second derivative test is inconclusive.

Notice that the First Derivative Test does not require the second derivative at all. You only need the sign of the first derivative on the intervals surrounding the critical point.

Common Mistakes

Mistake 1: Assuming every critical number is an extremum

A critical number is only a candidate for a local extremum. You must check whether the derivative changes sign. If the derivative is positive on both sides or negative on both sides, there is no local maximum or minimum.

Mistake 2: Checking only whether the derivative equals zero

The First Derivative Test is about the sign of the derivative, not merely whether it equals zero. A zero derivative tells you where to investigate; the sign change tells you what happens there.

Mistake 3: Forgetting points where the derivative is undefined

A critical number can occur when does not exist, as long as the original function is defined at that input. Sharp corners and cusps are common examples.

Mistake 4: Confusing local extrema with absolute extrema

A local maximum is higher than nearby function values, not necessarily every value in the entire domain. Similarly, a local minimum only needs to be lower than nearby values. Absolute extrema require comparing values across the entire relevant domain.

Practice Problems

Question 1

Use the First Derivative Test to find the local extrema of .

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Question 2

Use the First Derivative Test to classify the critical point of .

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Question 3

Find the local maximum and local minimum of .

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Question 4 (Derivative Undefined)

Use the First Derivative Test to classify the critical point of .

Show answer

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What the First Derivative Test Is For

The Key Idea

The First Derivative Test

Worked Examples

How to Read a Sign Chart

First Derivative Test vs. Second Derivative Test

Common Mistakes

Practice Problems

Test yourself: AP Calculus practice exams

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Get the Calc Pass App

Vol. 1: 1,114 Questions

Vol. 2: Solutions Manual

Related Tutorials

Using the First Derivative to Determine Increasing and Decreasing Intervals

Extreme Value Theorem and Candidates Test

The Second Derivative Test for Local Extrema

Finding Absolute Maximum and Minimum Values