Calculus I · 2B · lesson
The Second Derivative Test
Learn the second derivative test with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Theorems, extrema, and curve shapeWhat this section is building
Learn the second derivative test with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.
List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.
Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.
Learning objectives
Apply the Second Derivative Test and recognize inconclusive cases.
Classify Stationary Points with
Before the formulas
The theorem in The Second Derivative Test connects global information on an interval with local derivative behavior inside it. Read the hypotheses and conclusion separately. The theorem guarantees existence of at least one point; it may not identify the point, make it unique, or place it at the midpoint.
Use a diagram to understand the claim, then return to algebra to find candidate values when the problem asks for them. A correct theorem citation should name the interval and explain why each hypothesis is satisfied.
The second derivative classifies a horizontal tangent by local bending
At a critical point where , positive means the graph bends upward like a bowl, producing a local minimum. Negative means it bends downward, producing a local maximum.
If or does not exist, the test is inconclusive rather than false. Return to the first-derivative test or compare nearby values.
At a stationary point, measures whether the graph bends upward or downward. Positive second derivative suggests a local bowl and therefore a local minimum; negative second derivative suggests a cap and a local maximum.
When , the test is inconclusive, not evidence of "no extremum." The first derivative test or direct analysis must take over.
Second Derivative Test
Suppose .
• If , has a local minimum at . • If , has a local maximum at . • If , the test is inconclusive.
Classify two critical points
For
classify the critical points.
Worked solution
Write a real attempt before opening the supplied answer.
Inconclusive does not mean no extremum
For , and , but is a minimum. For , the same two derivative values occur, but is not an extremum. Use the First Derivative Test or another argument when .
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