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A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative
Statement
If is twice differentiable on an open interval, then is convex if and only if for every .
Facts & Assumptions
Given: A twice differentiable on an open interval .
A differentiable function on an open interval is convex if and only if its derivative is nondecreasing (A differentiable function on an open interval is convex if and only if its derivative is nondecreasing).
If a differentiable real function has nonnegative derivative on an interval, then it is nondecreasing on that interval (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
Assume is convex. By [L1], is nondecreasing; its difference quotients on either side of a point are nonnegative, and their common limit is therefore nonnegative.
Assume on . Applying [L2] to shows that is nondecreasing.
By [L1], the conclusion of step 1.2 makes convex, while step 1.1 proves the reverse implication.
Depends on
- A differentiable function on an open interval is convex if and only if its derivative is nondecreasing
- 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
- Higher derivatives and the classes $C^k$ and $C^\infty$
Used by
- A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there Corollary
- Positive-degree Dolbeault vanishing on pseudoconvex domains Corollary
- Squared distance is strictly convex along geodesics in a hadamard manifold Corollary
- No conjugate points in nonpositive constant curvature Example
- Power maps, endpoint distortion, and a non-Möbius quasisymmetric circle map Example
- The whole-line principal value of sin x / x is pi, so the half-line integral is pi / 2 Example
- Absolute real powers are Borel measurable and convex Lemma
- Backward uniqueness for the heat equation on a bounded interval Lemma
- Convexity, Lipschitz bounds, and oscillation of the source contrastive losses Lemma
- Finite-simplex entropy is continuous and concave Lemma
- Globally optimized constant-composition and Gallager functions agree Lemma
- Jordan's lemma for rational functions of one complex variable Lemma
- Negative entropy on the simplex is strongly convex and has range log n Lemma
- The finite KL generator has the required convex boundary extension Lemma
- The finite Rényi generators are convex with stated boundary values Lemma
- The negative reciprocal-power function is convex Lemma
- Water-filling maximizes the finite parallel Gaussian allocation objective Lemma
- A C² function is convex exactly when its Hessian is positive semidefinite Theorem
- Gallager random-coding exponent lower bound Theorem
- Kullback–Leibler divergence is nonnegative Theorem
- No conjugate points under nonpositive sectional curvature Theorem
Dependency tree · two levels
21 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
- R. Gardner, Convex Functions, Notes 6.6 (standard reference, not scraped)