C18 · Unit C5 Function analysis and optimization
Critical points and first-derivative evidence
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 5.3–5.5
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
Critical numbers are domain inputs where f minutes equals 0 or f minutes does not exist. They are candidates, not automatic extrema. A derivative changing from positive to negative gives a local maximum; negative to positive gives a local minimum, under continuity at the point. For absolute extrema on a closed interval, compare values at endpoints and all interior critical numbers.
Worked example
For f equals x cubed minus 3x on [ minus 2,3], f minutes equals 3(x minus 1)(x+1). Candidate values are f( minus 2) equals minus 2,f( minus 1) equals 2,f(1) equals minus 2,f(3) equals 18. The absolute minimum is minus 2 at both minus 2 and 1; maximum 18 at 3.
Explore the mathematics
Critical points and first-derivative evidence
Use derivative signs to classify stationary points and inflections.
Scroll sideways to see the full diagram.
Predict before continuing
Predict the stationary points and where concavity changes.
Read the complete explanation
- Predict the stationary points and where concavity changes.
- f′=0 at x=±1; use derivative sign changes to classify them.
- At −1 the function changes increasing to decreasing: a local maximum. At 1 it changes decreasing to increasing: a local minimum.
- Concavity changes at x=0 because f″=6x changes sign. A zero second derivative alone does not prove an inflection.
On [−2,3], compare both endpoints and critical points −1 and 1. The lowest value −2 occurs at both x=−2 and x=1. The largest value is f(3)=18.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
A stationary point with no sign change need not be an extremum: x cubed at zero is the standard example.
Investigate and explain
Build a sign chart with test inputs in every interval. Explain why checking f minutes only at critical numbers cannot classify extrema.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.