Concavity & 2nd Derivative Test

Unit 5 · Analytical Applications of Differentiation

Key Ideas

The sign of f′′ tells you how f is curving. At a critical point, f′′ can confirm whether the point is a local max or min — but only when f′′ ≠ 0.

Concavity

f′′(x) > 0 on (a, b) →

f \text{ is concave up on } (a,b) \text{ (curves like } \cup\text{)}

f′′(x) < 0 on (a, b) →

f \text{ is concave down on } (a,b) \text{ (curves like } \cap\text{)}

f′′ changes sign at c →

\text{inflection point at } c

Second Derivative Test — at critical point c where f′(c) = 0

f′′(c) > 0 →

\text{local minimum at } c

f′′(c) < 0 →

\text{local maximum at } c

f′′(c) = 0 →

\text{inconclusive — use First Derivative Test}

Caution: An inflection point requires that f′′ changes sign — f′′(c) = 0 alone is not sufficient. Example: f(x) = x⁴ has f′′(0) = 0 but no inflection point.

Type 1

Concavity & Inflection Points

The sign of f′′ determines whether f curves upward (concave up) or downward (concave down). An inflection point is where f′′ changes sign.

Example 1

Find the intervals of concavity and all inflection points.

f(x) = x^4 - 4x^3

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Type 2

2nd Derivative Test

At a critical point c where f′(c) = 0, check f′′(c): positive means local minimum, negative means local maximum, zero means inconclusive (use First Derivative Test instead).

Example 2

Find and classify all local extrema using the Second Derivative Test.

f(x) = x^3 - 3x

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