Using the Second Derivative Test to Determine Extrema
Apply the second derivative test at stationary points and use a different argument when it is inconclusive.
College Board Topic 5.7 · CED p. 100 (PDF p. 105) · CED effective Fall 2020; project course outline 2026–27.
At a stationary critical number c with f′(c) = 0, the second derivative test can classify the turn. Under the usual local differentiability conditions, f″(c) > 0 gives a local minimum and f″(c) < 0 gives a local maximum. The curve bends upward or downward around a horizontal tangent. First verify that the tangent is horizontal; concavity alone does not create an extremum.
If f″(c) = 0 or is unavailable, this test is inconclusive. Go back to derivative sign changes or direct comparison. The functions x⁴, −x⁴, and x³ all have first and second derivative zero at the origin, yet they give a minimum, maximum, and neither. One zero calculation cannot distinguish them.
Local and absolute conclusions still need different evidence. The CED also includes a useful one-critical-point case: when a continuous function on a domain interval has only one critical point, and that point is a local extremum, it is the corresponding absolute extremum on that interval. Continuity and the complete critical-point count matter. Any competing turn inside the interval would require another critical point. An equivalent practical proof is to show the function decreases all the way into a minimum and increases all the way away.
For f(x) = x + 4/x on (0,∞), f′ = 1 − 4/x² has only c = 2, and f″ = 8/x³ > 0. It is a local minimum; the unique-critical-point criterion makes it absolute. Directly, f′ is negative before 2 and positive after 2. On closed intervals with multiple candidates, use a complete candidates comparison instead.
Let's work through it
For f(x) = x³ − 12x, f′ = 3x² − 12 gives critical numbers −2 and 2.
Why: Solve the stationary-point condition before applying the test.
f″(x) = 6x; f″(−2) = −12 and f″(2) = 12.
Why: The second derivative supplies local bending at the stationary points.
There is a local maximum at −2 and local minimum at 2.
Why: Negative/positive second derivative at a stationary point classifies the local extremum.
These are not absolute extrema on all real numbers.
Why: The cubic tends to opposite infinities, and the one-critical-point condition is not satisfied.
Try it from memory
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Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
What must you check before using f″(c) to classify an extremum?
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Retrieval 1
What must you check before using f″(c) to classify an extremum?
Worked answer and reason
That c is a stationary critical number with f′(c) = 0 and the needed local differentiability conditions hold.
A tilted concave-up curve need not have a minimum at c.
Retrieval 2
What does f″(c) = 0 conclude?
Worked answer and reason
The test is inconclusive.
A first-derivative sign test can distinguish minimum, maximum, and neither.
Retrieval 3
Can one local minimum be absolute on an open interval?
Worked answer and reason
Yes, for a continuous function with exactly one critical point there, as in x + 4/x on (0,∞).
The complete interval behavior prevents another competing turn.
Catch the tempting mistake
- The second derivative zero means neither maximum nor minimum.
- It means this test cannot decide.
- Local minimum automatically means absolute minimum.
- You need additional interval evidence, such as a complete candidate comparison or the one-critical-point criterion.
Calculator workspace and limits
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Put it to work
- Unit 5 problem 14 · No calculator · analytic
- Unit 5 problem 15 · No calculator · analytic
- Unit 5 problem 16 · No calculator · analytic
For the teacher
- Make “inconclusive” an acceptable deliberate answer, followed by a different valid test.
- Ask learners to state the interval and account for all critical points before using the unique-critical-point conclusion.
Glossary and deeper reading · 8 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Critical point
A critical number is an input in the function’s domain where its derivative is zero or does not exist. The corresponding point on the graph is called a critical point. For example, x³ has a critical number at 0, but it has no local maximum or minimum there. Critical numbers are candidates for extrema and need further analysis; inputs outside the domain are excluded.
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Differentiability
Differentiability means that a function has a finite derivative at the point under discussion. Differentiability at a point implies continuity there, but continuity alone does not imply differentiability. For example, |x| is continuous at 0 but its left and right tangent slopes differ, so it is not differentiable there. Corners, cusps, vertical tangents, and discontinuities can prevent a finite derivative.
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Derivative
A derivative gives the instantaneous rate of change of a function with respect to its input. At a point, it is the finite limit of the difference quotient when that limit exists. For example, the derivative of x² is 2x, so the tangent slope at x=3 is 6. A derivative function collects those rates across the inputs where the function is differentiable.
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Function
A mathematical function assigns exactly one output to each input in its domain. Inputs can share an output, but one input cannot have two different outputs. For example, f(x)=x² assigns the output 9 to both 3 and −3. Function notation names the rule and its input; f(3) means the output when the input is 3.
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Continuity
A function is continuous at an input when its value exists, its finite limit exists, and those two values agree. At a domain endpoint, continuity uses the relevant one-sided approach. For example, x² is continuous at 3 because both its value and its nearby limit are 9. Continuity on an interval requires the appropriate condition at every input in that interval.
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Domain
The domain of a function is the set of inputs for which the function is defined. An expression and its context can both restrict the allowed inputs. For example, the real function f(x)=1/(x−2) has domain all real numbers except 2. For a model of elapsed time, a sensible domain may also require t≥0.
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Local extremum
A local extremum is a value at least as large as, or at least as small as, all values at nearby allowed inputs. A local maximum compares nearby heights above, while a local minimum compares them below. For example, x² has a local minimum at 0. A local extremum need not be the greatest or least value over the entire domain.
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Interval
An interval is a set of real numbers containing every number between any two of its members. Parentheses exclude an endpoint, while square brackets include it. For example, (1,4] includes 4 and every number between 1 and 4, but excludes 1. An infinite endpoint always uses a parenthesis because infinity is not a real endpoint.