{"schema_version":"ocme-site-results-v1","locale":"en","page_type":"derivations","facts":{"questions":36000,"reviewed":36000,"display_candidates":0,"knowledge":302,"guides":299,"paths":59,"batches":360,"families":1441,"atlas":302,"core":80,"evidence":9,"formal_evidence":5,"formal_subjects":4,"lean_files":6,"lean_toolchain":"leanprover/lean4:v4.30.0","question_state":"candidate","english_state":"ai_candidate","question_Lean":"NOT_RUN","canonical_truth_promotion":false},"filters":{"q":"mko-convex-function"},"kind":"guides","total":1,"pages":1,"page":1,"entries":[{"id":"derivation-mko-convex-function-01","title":{"zh-Hant":"凸函數：二次函數 Jensen gap 的完整證書與單一 Jensen 反例","en":"Convex function: a complete certificate via the Jensen gap of a quadratic, and a single Jensen counterexample"},"summary":{"zh-Hant":"（一）證明 f(x)=2x²−3x+1 在區間 I=[−1,4] 上是凸函數：先確認 I 是凸集合，再對所有 x,y∈I 與所有 λ∈[0,1] 證明 f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y)，方法是證明二次函數的 Jensen gap 等於 aλ(1−λ)(x−y)²。（二）由同一恆等式得仿射函數 ℓ(x)=3x−5 在 I 上取等號。（三）用一個精確的 Jensen 違反否證 g(x)=−x²+x 在 J=[0,2] 上的凸性，並說明幾個通過的樣本為何不能證明凸性。","en":"(1) Prove that f(x)=2x²−3x+1 is convex on the interval I=[−1,4]: first confirm that I is convex, then prove f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y) for all x,y∈I and all λ∈[0,1], by showing that the Jensen gap of a quadratic equals aλ(1−λ)(x−y)². (2) From the same identity, the affine function ℓ(x)=3x−5 gives equality on I. (3) Refute the convexity of g(x)=−x²+x on J=[0,2] with one exact Jensen violation, and explain why a few passing samples cannot prove convexity."},"mko_ids":["mko-convex-function"]}],"title":"Concept derivation guides"}