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The second-derivative test for strict local extrema
Statement
Suppose and exists and is continuous near . If , then is a strict local minimum; if , then is a strict local maximum.
Facts & Assumptions
Given: The hypotheses at .
Continuity preserves a strict sign on a sufficiently small neighbourhood (Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point).
A positive derivative gives strict increase and a negative derivative gives strict decrease (On an interval , for continuous on and differentiable at every interior point: throughout gives nondecreasing, gives increasing, and give the two decreasing forms; conversely a nondecreasing has and a nonincreasing has wherever it is differentiable, and no strict converse is claimed).
Proof
If , [L1] gives an interval about on which . Hence is strictly increasing there; since , to the left and to the right.
If , apply the preceding argument to ; this gives a strict local maximum.
Applying [L2] to , it decreases toward from the left and increases away from on the right, so is a strict local minimum.
The two stated sign cases are exhausted.
Depends on
- Higher derivatives and the classes $C^k$ and $C^\infty$
- On an interval $I$, for $f$ continuous on $I$ and differentiable at every interior point: $f' \ge 0$ throughout gives $f$ nondecreasing, $f' > 0$ gives $f$ increasing, $f' \le 0$ and $f' < 0$ give the two decreasing forms; conversely a nondecreasing $f$ has $f' \ge 0$ and a nonincreasing $f$ has $f' \le 0$ wherever it is differentiable, and no strict converse is claimed
- Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
- The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$, and differentiability on a set
Used by
Dependency tree · two levels
22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)