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Concavitas, convexitas et pvncta inflexionis

Svmmarivm

Haec lectio investigat qvomodo delineatio fvnctionis flectatvr: si inter dvo eivs pvncta infra chordam iacet qvae ea ivngit, fvnctio convexa est; si svpra eam iacet, concava. Definitionem in comparationem declivitatvm chordarvm vertit et, theoremate valoris medii et monotonia adhibitis, ostendit fvnctionem derivabilem convexam esse si et solvm si derivata eivs crescens est, si et solvm si delineatio eivs svpra omnes tangentes svas iacet, et fvnctionem bis derivabilem convexam esse si et solvm si derivata eivs secvnda negativa non est. Pvncta inflexionis definit, vbi fvnctio ex convexa in concavam transit, et criteria dat qvibvs invenivntvr: mvtationem signi derivatae secvndae, condicionem necessariam f′′(a)=0f''(a) = 0 et criterivm derivatae tertiae, qvod delineationem cvm tangente sva comparat sicvt criterivm derivatarvm svccessivarvm eam cvm recta horizontali comparavit. Chordis sinvs inaeqvalitatem Iordani obtinet, et inaeqvalitate Ienseniana concavitatem logarithmi in inaeqvalitatem inter medias geometricam et arithmeticam qvotlibet nvmerorvm convertit.

Proposita discendi

  1. Fvnctionem convexam et concavam per chordas earvm cvm qvantificatoribvs definire, dvas conventiones librorvm distingvere et definitionem in comparationem declivitatvm chordarvm vertere.
  2. Ostendere fvnctionem derivabilem convexam esse si et solvm si derivata eivs crescens est et si et solvm si delineatio eivs svpra tangentes svas iacet, et derivata secvnda decernere in qvibvs intervallis fvnctio convexa avt concava sit.
  3. Pvncta inflexionis fvnctionis mvtatione signi derivatae secvndae et derivata tertia invenire, et pvncta cvm derivata secvnda nvlla agnoscere qvae pvncta inflexionis non svnt.
  4. Inaeqvalitates convexitate ostendere: Iordani per chordas, Ienseniana et inaeqvalitatem mediarvm.

Fvnctiones convexae et concavae

Lectio de monotonia investigavit vtrvm fvnctio ascendat an descendat; haec investigat qvomodo id faciat. Fvnctiones x2x^2 et x\sqrt{x} stricte crescentes svnt in [0,∞)\ico{0}{\infty}, sed prima svrsvm, altera deorsvm flectitvr: segmentvm qvod dvo pvncta delineationis ipsivs x2x^2 ivngit, chorda eivs, svpra arcvm delineationis inter ea iacet, et segmentvm ipsivs x\sqrt{x} infra evm. Si tt in [0,1]\icc{0}{1} est, pvnctvm chordae qvae (x,f(x))(x, f(x)) cvm (y,f(y))(y, f(y)) ivngit, cvivs abscissa est tx+(1−t)ytx + (1 - t)y, ordinatam tf(x)+(1−t)f(y)t f(x) + (1 - t) f(y) habet: eandem combinationem ordinatarvm.

DefinitioFvnctio convexa et fvnctio concava

Sit ff fvnctio in intervallo II definita. Dicitvr ff convexa in II si nvllvm pvnctvm delineationis eivs svpra chordam iacet qvae dvo pvncta delineationis ivngit inter qvorvm abscissas eivs abscissa est:

(∀x∈I)(∀y∈I)(∀t∈[0,1])(f(tx+(1−t)y)≤tf(x)+(1−t)f(y))(\forall x \in I)(\forall y \in I)(\forall t \in \icc{0}{1})\left(f\left(tx + (1 - t)y\right) \leq t f(x) + (1 - t) f(y)\right)

et stricte convexa si inaeqvalitas stricta est qvotiens x≠yx \neq y et 0<t<10 < t < 1. Dicitvr ff concava (vel stricte concava) in II si −f-f convexa (vel stricte convexa) est, id est, si inaeqvalitas opposita valet: nvllvm pvnctvm delineationis infra eas chordas iacet.

Fvnctio linearis mx+nmx + n simvl convexa et concava est: delineatio eivs cvm chordis svis congrvit. Fvnctio x2x^2 stricte convexa est in R\mathbb{R}, qvia differentia inter dvo membra inaeqvalitatis est t(1−t)(x−y)2t(1 - t)(x - y)^2, positiva si x≠yx \neq y et 0<t<10 < t < 1. Valor absolvtvs convexvs est, per inaeqvalitatem triangvlarem, ∣tx+(1−t)y∣≤t∣x∣+(1−t)∣y∣\abs{tx + (1 - t)y} \leq t\abs{x} + (1 - t)\abs{y}, qvamqvam in 00 derivabilis non est; stricte convexvs non est, qvia in [0,∞)\ico{0}{\infty} cvm chordis svis congrvit.

Dvae conventiones. In hoc cvrsv «convexa» est fvnctio cvivs delineatio svpra chordas svas non ascendit, sicvt x2x^2, et «concava» ea qvae infra eas non descendit, sicvt x\sqrt{x}. Haec est conventio plvrimorvm librorvm analyseos et optimizationis, atqve ea facti admissi de concavitate sinvs, in lectione de fvnctionibvs trigonometricis earvmqve delineationibvs, ex Algebra et Geometria II. Alii libri verbis contrario modo vtvntvr, et x2x^2 concavam vocant, qvia delineationem ex svperiore loco aspicivnt; alii «concava svrsvm» pro convexa et «concava deorsvm» pro concava dicvnt. Novo libro oblato, convenit definitionem eivs qvaerere vel animadvertere qvomodo x2x^2 appellet: inaeqvalitates non mvtantvr, nomina tantvm.

Definitio pvnctvm delineationis cvm chorda comparat. Si x<z<yx < z < y, comparare (z,f(z))(z, f(z)) cvm chorda a xx ad yy idem valet ac comparare declivitates trivm chordarvm: dvm delineatio fvnctionis convexae a sinistra ad dextram percvrritvr, chordae ardviores fivnt. Figvra id ostendit cvm f(x)=x24−0,3x+0,6f(x) = \frac{x^2}{4} - 0{,}3x + 0{,}6.

LemmaDeclivitates chordarvm

Sit ff fvnctio in intervallo II definita, et sit S(u,v)=f(v)−f(u)v−uS(u, v) = \frac{f(v) - f(u)}{v - u} declivitas chordae inter dvas abscissas distinctas uu et vv ipsivs II. Si x<z<yx < z < y pvncta ipsivs II svnt, vnaqvaeqve inaeqvalitatvm

S(x,z)≤S(x,y),S(x,y)≤S(z,y),S(x,z)≤S(z,y)S(x, z) \leq S(x, y), \qquad S(x, y) \leq S(z, y), \qquad S(x, z) \leq S(z, y)

idem valet ac pvnctvm (z,f(z))(z, f(z)) non iacere svpra chordam qvae (x,f(x))(x, f(x)) cvm (y,f(y))(y, f(y)) ivngit, et vnaqvaeqve inaeqvalitatvm strictarvm, id stricte infra eam iacere. Itaqve ff convexa est in II si et solvm si tres inaeqvalitates valent pro omnibvs x<z<yx < z < y ipsivs II, et stricte convexa si et solvm si stricte valent.

Demonstratio

  1. f(z)≤tf(x)+(1−t)f(y)⇔(y−x)f(z)≤(y−z)f(x)+(z−x)f(y)f(z) \leq t f(x) + (1 - t) f(y) \Leftrightarrow \dato{c}{(y - x) f(z) \leq (y - z) f(x) + (z - x) f(y)}

    Sit t=y−zy−xt = \frac{y - z}{y - x}, qvod in (0,1)\ioo{0}{1} est; tvm 1−t=z−xy−x1 - t = \frac{z - x}{y - x} et z=tx+(1−t)yz = tx + (1 - t)y. Pvnctvm chordae abscissae zz ordinatam tf(x)+(1−t)f(y)t f(x) + (1 - t) f(y) habet, et comparatio, per y−x>0y - x > 0 mvltiplicata, sine fractionibvs scribitvr:

  2. (y−x)(f(z)−f(x))≤(z−x)(f(y)−f(x))⇔S(x,z)≤S(x,y)(y - x)\left(f(z) - f(x)\right) \leq (z - x)\left(f(y) - f(x)\right) \Leftrightarrow \resaltar{S(x, z) \leq S(x, y)}

    Ab vtroqve membro (y−x)f(x)(y - x) f(x) svbtrahitvr; cvm y−x=(y−z)+(z−x)y - x = (y - z) + (z - x), membrvm altervm fit (z−x)(f(y)−f(x))(z - x)\left(f(y) - f(x)\right), et satis est per (z−x)(y−x)>0(z - x)(y - x) > 0 dividere. Eodem modo, (y−x)f(y)(y - x) f(y) svbtracto et divisione per (y−z)(y−x)>0(y - z)(y - x) > 0 facta, altera inaeqvalitas obtinetvr.

  3. S(x,y)=z−xy−xS(x,z)+y−zy−xS(z,y),S(x,y)−S(x,z)=y−zy−x(S(z,y)−S(x,z))S(x, y) = \frac{z - x}{y - x} S(x, z) + \frac{y - z}{y - x} S(z, y), \qquad S(x, y) - S(x, z) = \resaltar{\frac{y - z}{y - x}\left(S(z, y) - S(x, z)\right)}

    Incrementvm a xx ad yy est svmma incrementorvm a xx ad zz et a zz ad yy: declivitas chordae longae media est declivitatvm brevivm, ponderibvs positivis svmmae 11, et differentia eivs a S(x,z)S(x, z) signvm ipsivs S(z,y)−S(x,z)S(z, y) - S(x, z) habet. Ideo tres inaeqvalitates aeqvivalent, et similiter strictae. Qvod ad definitionem attinet, si x=yx = y, vel si tt valet 00 avt 11, inaeqvalitas cvm aeqvalitate valet, et si x>yx > y permvtantvr mvnera ipsorvm xx et yy atqve ipsorvm tt et 1−t1 - t.