Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Every slope between the left and right derivatives of a convex function gives a supporting line

Statement

Let f:I→R be convex on an open interval, let c∈I, and let m satisfy f−′(c)≤m≤f+′(c). Then the line x↦f(c)+m(x−c) supports f at c.

Facts & Assumptions

Given: A convex f:I→R, an interior point c, and f−′(c)≤m≤f+′(c).

[L1]

For u<v, f−′(u)≤f+′(u)≤(f(v)−f(u))/(v−u)≤f−′(v)≤f+′(v) (A convex function on an open interval has finite left and right derivatives everywhere, with f−′(u)≤f+′(u)≤(f(v)−f(u))/(v−u)≤f−′(v)≤f+′(v) for u<v).

[L2]

A line of slope m supports f at c when f(x)≥f(c)+m(x−c) throughout the interval (A supporting line of slope m for a real function at an interior point).

Proof

technique · direct
1.1L1L2

If x<c, [L1] applied to x<c gives (f(c)−f(x))/(c−x)≤f−′(c)≤m.

2.1step 1.1L2algebra

If x>c, [L1] gives m≤f+′(c)≤(f(x)−f(c))/(x−c).

3.1step 1.1step 2.1∎

Multiplying the inequalities in steps 1.1 and 2.1 by their positive denominators and rearranging yields f(x)≥f(c)+m(x−c) on both sides of c, while equality holds at c; hence [L2] applies.

Depends on

Dependency tree · 2 levels

Sources