Using the First Derivative Test to Determine Relative (Local) Extrema
Use one-sided derivative sign changes to classify a local extremum, including a continuous corner.
College Board Topic 5.4 · CED p. 97 (PDF p. 102) · CED effective Fall 2020; project course outline 2026–27.
At a critical number c, look at f′ on both sides. A change from positive to negative means f climbs into c and falls away: a local maximum. A change from negative to positive means f falls into c and climbs away: a local minimum. If the sign does not change, this test finds neither. State the sign change, not merely the fact that the derivative is zero.
The first derivative test works at a corner too, provided f is continuous at the point and differentiable on suitable nearby sides. For |x|, the derivative changes from −1 to +1 at zero, so there is a local minimum even though f′(0) does not exist. Do not transfer this conclusion across a jump: nearby monotone pieces do not control the value of a disconnected point.
For f′(x) = x²(x − 3), the derivative is negative on both sides of zero but switches from negative to positive at 3. The critical number zero is neither a local maximum nor a minimum; 3 is a local minimum. A complete justification names f, the location, the derivative signs, and the conclusion.
Let's work through it
Let f(x) = x⁴ − 4x³. Then f′(x) = 4x²(x − 3).
Why: The derivative zeros are 0 and 3.
At zero, f′ stays negative on both sides; there is no extremum.
Why: The function continues decreasing through a horizontal tangent.
At 3, f′ changes from negative to positive; f has a local minimum there.
Why: A fall followed by a rise is the first derivative test for a local minimum.
The local minimum value is f(3) = 81 − 108 = −27.
Why: Evaluate the original function for the output value, not its derivative.
Try it from memory
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Question 1
What sign change proves a local maximum?
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Retrieval 1
What sign change proves a local maximum?
Worked answer and reason
f′ changes from positive to negative.
The function rises before and falls after the point.
Retrieval 2
If f′ is positive on both sides of a critical number, what happens?
Worked answer and reason
No local extremum is established by the first derivative test.
The function keeps increasing.
Catch the tempting mistake
- Every zero of f′ is a maximum or minimum.
- A sign change is needed; x³ supplies a horizontal tangent without an extremum.
- A derivative maximum is also a maximum of f.
- A maximum of f′ concerns f’s slope; an extremum of f concerns a derivative sign change.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
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Put it to work
- Unit 5 problem 8 · No calculator · analytic
- Unit 5 problem 9 · No calculator · graphical
- Unit 5 problem 29 · No calculator · multiple
For the teacher
- Require one complete sign-change sentence after each classification.
- Include a touch-zero example and a corner example in the same exit ticket.
Glossary and deeper reading · 4 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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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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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.