Calculus I · Unit 2B · lesson

The Second Derivative Test

Concept

Learning objectives

Apply the Second Derivative Test and recognize inconclusive cases.

Classify Stationary Points with $f"$

Explanation

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.

Explanation

The second derivative classifies a horizontal tangent by local bending

At a critical point where $f'=0$ , positive $f"$ means the graph bends upward like a bowl, producing a local minimum. Negative $f"$ means it bends downward, producing a local maximum.

If $f"=0$ 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, $f"$ 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 $f"=0$ , the test is inconclusive, not evidence of "no extremum." The first derivative test or direct analysis must take over.

Theorem

Second Derivative Test

Suppose $f'(c)=0$ .

• If $f"(c)>0$ , $f$ has a local minimum at $c$ . • If $f"(c)<0$ , $f$ has a local maximum at $c$ . • If $f"(c)=0$ , the test is inconclusive.

Guided walkthrough

Classify two critical points

For

f(x)=x^3-3x,

classify the critical points.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

Inconclusive does not mean no extremum

For $f(x)=x^4$ , $f'(0)=0$ and $f"(0)=0$ , but $0$ is a minimum. For $g(x)=x^3$ , the same two derivative values occur, but $0$ is not an extremum. Use the First Derivative Test or another argument when $f"(c)=0$ .

After the explanation

Use the section idea

Reading lens

Turn derivative signs and theorem hypotheses into a defensible account of extrema, monotonicity, concavity, and global shape.

Mental model

Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.

Decision

List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.

Common trap

Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.

Check yourself

Can every turn, bend, endpoint result, and asymptote in your sketch be traced to algebraic evidence?

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Classify Stationary Points with f"f"f"

Original instruction with traceable references.

Classify Stationary Points with $f"$