Connecting a Function, Its First Derivative, and Its Second Derivative
Connect heights and slopes across a function and its first two derivatives without conflating their outputs.
College Board Topic 5.9 · CED p. 102 (PDF p. 107) · CED effective Fall 2020; project course outline 2026–27.
Read the chain of relationships one level at a time. A height on f′ is a slope on f. A height on f″ is a slope on f′. Consequently, f′ > 0 and f″ < 0 mean f increases while its slopes decrease: increasing and concave down. Neither inequality says whether f is above zero.
To match three graphs, look for landmarks that survive this chain. Extrema of f usually align with sign-changing zeros of f′. Extrema of f′ often align with sign-changing zeros of f″ and thus inflections of f, when continuity and domain conditions hold. A non-sign-changing zero can fail to make an extremum, and an undefined point demands a separate check. Use several features, not one coincidence.
For f(x) = x³, f′ = 3x² is nonnegative and touches zero at the origin, while f″ = 6x crosses zero there. The function increases through its horizontal tangent and changes concavity at zero. That one example keeps a stationary inflection distinct from a maximum or minimum. Tables can show the same connections: if exact f′ entries are positive but shrink, they are consistent with increase and concavity down, though isolated entries alone do not prove interval behavior.
Let's work through it
For f(x) = −x² + 4x + 1, f′(x) = −2x + 4 and f″(x) = −2.
Why: Differentiate once and twice to expose the chain.
The f′ graph crosses from positive to negative at x = 2.
Why: Its zero and sign change identify a local maximum of f.
The f″ graph is below the axis everywhere, so f′ decreases and f is concave down.
Why: Second derivative height supplies first derivative slope.
f(2) = 5 is the maximum height of f, while f′(2) = 0 is its tangent slope.
Why: Matching x-coordinates does not make the y-values of different graphs equal.
Try it from memory
Pause & retrieve
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
If f′ > 0 and f″ > 0, how does f behave?
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
If f′ > 0 and f″ > 0, how does f behave?
Worked answer and reason
Increasing and concave up.
The first inequality controls slope sign; the second controls slope change.
Retrieval 2
A local maximum of f′ with a derivative sign change in f″ usually corresponds to what feature of f?
Worked answer and reason
An inflection point, if the function is continuous there and concavity changes.
An extremum of slope is a change in how slopes evolve.
Catch the tempting mistake
- f″ < 0 means f is decreasing.
- It means f′ decreases; f itself could be increasing with a shrinking positive slope.
- A zero in any one of the three graphs means all three are zero there.
- Different graphs measure different quantities at the same input.
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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+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 5 problem 18 · No calculator · graphical
- Unit 5 problem 19 · No calculator · verbal
- Unit 5 problem 20 · No calculator · numerical
- Unit 5 problem 29 · No calculator · multiple
For the teacher
- Have students explain a graph match using at least two independent landmarks.
- Use a verbal card such as “negative, increasing, concave down” and ask which parts a derivative graph alone can establish.
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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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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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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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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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.