C19 · Unit C5 Function analysis and optimization
Concavity, inflection and curve reconstruction
CoreReview editionNot yet human-reviewed
- Time
- About 45 minutes, untimed
- Course framework reference
- AP 5.6–5.9, 5.12
- Builds on
- Part of
Progress on this lesson counts in every pathway listed.
Understand the idea
Concavity concerns whether slopes increase or decrease: f seconds >0 gives concave up, f seconds <0 concave down. An inflection point requires a change of concavity at a point on the graph. If f minutes (c) equals 0 and f seconds (c)>0, the second-derivative test gives a local minimum; f seconds (c)<0 gives a maximum. A zero second derivative makes this test inconclusive. Apply these ideas to implicit curves on valid local branches.
Worked example
For f equals x to the power 4, f seconds equals 12x squared is greater than or equal to 0, so zero is not an inflection even though f seconds (0) equals 0. For f equals x cubed, f seconds equals 6x changes sign at zero, so (0,0) is an inflection. The derivative test and concavity test answer different questions.
Explore the mathematics
Concavity, inflection and curve reconstruction
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.
The nonnegative curve x⁴ is concave up on both sides of zero. The curve x³ changes concavity there. A zero second derivative alone does not establish an inflection point.
Discuss: explain what each label represents, then draw a related example with different values.
A common misconception
Increasing does not imply concave up. A function can rise while its positive slope becomes smaller.
Investigate and explain
Sketch f from a qualitative graph of f minutes. Mark increasing intervals and inflections using the sign and trend of f minutes separately.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.