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′′(x) < 0 on (a, b) →
f′′ changes sign at c →
Second Derivative Test — at critical point where
f′′(c) > 0 →
f′′(c) < 0 →
f′′(c) = 0 →
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.
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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.
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