Determining Intervals on Which a Function Is Increasing or Decreasing
Use a complete derivative sign chart to justify increasing and decreasing intervals.
College Board Topic 5.3 · CED p. 96 (PDF p. 101) · CED effective Fall 2020; project course outline 2026–27.
The sign of f′ tells you the direction of f’s change. Where f′ > 0 on an interval, f is increasing; where f′ < 0, f is decreasing. This can be justified by the Mean Value Theorem between any two points in that interval. You do not need f to be above the horizontal axis to be increasing. A negative-valued function can still climb.
Make a sign chart by splitting the domain at derivative zeros, points where the derivative is undefined, and domain breaks. Determine the sign on each open piece. Test points work when continuity and the complete list of zeros/breaks ensure the sign cannot change within that piece. A random numerical sample by itself does not prove an interval claim.
For f′(x) = (x + 1)(x − 2), both factors are negative on x < −1, their signs differ between −1 and 2, and both are positive for x > 2. Thus f increases, decreases, then increases. On a derivative graph, those signs are its heights above or below the axis. The graph’s slope is a different piece of information, saved for concavity.
| Interval | (−∞,−1) | (−1,2) | (2,∞) |
|---|---|---|---|
| Derivative sign | + | − | + |
An isolated zero derivative does not force a flat interval. The function x³ is strictly increasing through zero even though f′(0) = 0. If f′ is zero throughout an interval, the MVT instead shows f is constant there.
Let's work through it
Let f(x) = x³ − 3x. Then f′(x) = 3(x − 1)(x + 1).
Why: Differentiation provides a factored expression suitable for a sign chart.
The derivative is positive on (−∞,−1) and (1,∞), negative on (−1,1).
Why: Check factor signs between the complete set of zeros.
f increases on the positive-sign intervals and decreases on the negative-sign interval.
Why: The derivative sign controls the direction of function change.
No claim about whether f itself is positive follows from these signs alone.
Why: Increasing/decreasing is about relative values, not location above the axis.
Try it from memory
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Try it before you peek.
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
A derivative graph lies below the axis throughout an interval. What does f do?
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Retrieval 1
A derivative graph lies below the axis throughout an interval. What does f do?
Worked answer and reason
It decreases on that interval.
Derivative height is the signed slope of f.
Retrieval 2
Does f′(c) = 0 alone make f constant near c?
Worked answer and reason
No.
An isolated zero differs from a derivative that vanishes throughout an interval.
Catch the tempting mistake
- To find where f increases, check where f > 0.
- Check f′ > 0; function height and direction of change are independent.
- One positive numerical derivative proves increase on the whole interval.
- It establishes evidence at one point; an interval conclusion needs interval evidence.
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 6 · No calculator · analytic
- Unit 5 problem 7 · No calculator · numerical
- Unit 5 problem 29 · No calculator · multiple
For the teacher
- Use separate colors for a derivative graph’s height and slope.
- When a learner tests one point, ask why no unlisted derivative zero or discontinuity could intervene.
Glossary and deeper reading · 6 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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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.
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Mean Value Theorem
The Mean Value Theorem connects an average rate on a closed interval to an instantaneous rate at an interior point. It requires continuity on the closed interval and differentiability on its interior. For example, f(x)=x² on [1,3] has average slope 4 and derivative 4 at x=2. The theorem guarantees at least one such interior point but need not identify a unique one.
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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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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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Concavity
Concavity describes how a graph’s slope changes as the input increases. A graph is concave up where its derivative increases and concave down where its derivative decreases. For example, x² is concave up because its derivative 2x increases. A positive second derivative establishes concave-up behavior on an interval, while a negative one establishes concave-down behavior.
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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.